From f81e9183951ed4871a790993f82d6b1fb6f091cb Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Mon, 17 Aug 2026 21:58:21 +0900 Subject: [PATCH 01/12] Stop keeping a second copy of GeometricOptimizers' geometry MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Eleven types were defined in both packages -- `Manifold`, `StiefelManifold`, `GrassmannManifold`, `SkewSymMatrix`, `SymmetricMatrix`, `AbstractTriangular`, `LowerTriangular`, `UpperTriangular`, `AbstractLieAlgHorMatrix`, `StiefelLieAlgHorMatrix`, `GrassmannLieAlgHorMatrix`, `StiefelProjection` -- as near-verbatim copies. Julia saw them as distinct types, so none of GeometricOptimizers' generic machinery dispatched on GML's: `geodesic(::Manifold, ::AbstractMatrix)`, `cayley`, `global_rep`, `apply_section!` and `update_section!` all had to be written again per manifold, and `src/optimizers/go_bridges.jl` existed to reconnect the two hierarchies with about thirty more methods. They are `import`ed now and the copies are gone: 2595 deletions against 202 insertions. `import` and not `const X = GeometricOptimizers.X` -- with `using GeometricOptimizers: X`, every constructor GML adds to an imported type warns ("Constructor for type X was extended without explicit qualification"), and an `import` is also where a reader looks to find out where a name comes from. This is issue B2, and it needed GeometricOptimizers#50 first: `SymmetricMatrix` and the triangular types had no `similar`, `fill!` or elementwise primitives there, so they could not have been optimizer parameters. ## Three bugs that the duplication was hiding Each was live in GML's copy and already fixed in GeometricOptimizers', so importing fixes them. All three were reproduced before and after. - **`rand(3,3) + SkewSymMatrix(...)` was a `StackOverflowError.`** `Base.:+(B::AbstractMatrix, A::SkewSymMatrix) = B + A` called itself. Upstream has `= A + B`. - **`parent(::StiefelLieAlgHorMatrix)` returned an unbound `B`.** It read `(A, B)` where it meant `(A.A, A.B)`; upstream's `vec(::AbstractLieAlgHorMatrix)` is built on `parent`, so the two disagreed. - **A decaying step size was read one iteration early.** `optimization_step!` read the step size *before* incrementing `opt.iterations`, so a run's first step was `α(0) = η₁` where the pre-0.5 code incremented first and took `α(1) = γη₁`. Every step of a run sat one place early in the schedule. Upstream's `test/adam_optimizer_with_decay.jl` asserts the behaviour this restores. ## One `AdamOptimizerWithDecay` (issue B1) Both packages exported the name, so `using GeometricMachineLearning, GeometricOptimizers` was an `UndefVarError` on it. They were the same algorithm -- Adam's direction with a geometrically decaying learning rate, the same `γ = exp(log(η₂/η₁)/n)`, values identical to the last bit -- differing only in packaging: a `struct <: OptimizerMethod` carrying both halves here, an `(algorithm, linesearch)` pairing upstream, where the step size belongs to a `LinesearchMethod`. GML's struct is deleted and upstream's name imported. `Optimizer` takes the pairing by splatting, as upstream's does: Optimizer(nn; AdamOptimizerWithDecay(n_epochs, Float32)...) `step_size` accepts a `DecayingStatic` as well as a number, and `_current_step_size` asks the schedule. The `AdamOptimizerWithDecay` *methods* of `_is_go_native_method`, `_adapt_method_to_T`, `_default_step_size` and `_current_step_size` go with the struct -- the functions themselves stay, they still dispatch on the upstream methods -- and so does a `_euclidean_update!` that was a verbatim copy of the `Adam` one. `Optimizer` is now the only name both packages export -- twelve did before -- and it stays until the parameter-tree traversal moves upstream too. See C1. ## What stayed `src/arrays/gml_extensions.jl` holds the methods that are genuinely about neural networks rather than geometry: `add!` (AbstractNeuralNetworks') and `networkbackend` for the imported types. `PoissonTensor` stays where it was. The tensor kernels and their `rrule`s keep dispatching on the imported types, which is what they always did -- only the module the types come from changed. `update!` and `solve!` stopped being exported. Neither was GML's: `update!` had zero methods here and the export only shadowed upstream's, which is the one that has methods for the caches; `solve!` was a second generic function of the same name. Both were export collisions of exactly B1's kind. Co-Authored-By: Claude Opus 5 (1M context) --- src/GeometricMachineLearning.jl | 113 ++++-- .../transformer_neural_network.jl | 2 +- src/arrays/abstract_lie_algebra_horizontal.jl | 8 - src/arrays/gml_extensions.jl | 43 ++ .../grassmann_lie_algebra_horizontal.jl | 209 ---------- src/arrays/lower_triangular.jl | 96 ----- src/arrays/skew_symmetric.jl | 357 ---------------- src/arrays/stiefel_lie_algebra_horizontal.jl | 329 --------------- src/arrays/stiefel_projection.jl | 84 ---- src/arrays/symmetric.jl | 310 -------------- src/arrays/triangular.jl | 164 -------- src/arrays/upper_triangular.jl | 105 ----- src/layers/linear_symplectic_attention.jl | 2 +- src/layers/multi_head_attention.jl | 2 +- src/layers/sympnets.jl | 6 +- src/layers/volume_preserving_feedforward.jl | 2 +- src/manifolds/abstract_manifold.jl | 116 ------ src/manifolds/grassmann_manifold.jl | 236 ----------- src/manifolds/stiefel_manifold.jl | 380 ------------------ src/optimizers/go_bridges.jl | 108 ----- src/optimizers/optimizer.jl | 119 ++++-- src/utils.jl | 6 +- 22 files changed, 202 insertions(+), 2595 deletions(-) delete mode 100644 src/arrays/abstract_lie_algebra_horizontal.jl create mode 100644 src/arrays/gml_extensions.jl delete mode 100644 src/arrays/grassmann_lie_algebra_horizontal.jl delete mode 100644 src/arrays/lower_triangular.jl delete mode 100644 src/arrays/skew_symmetric.jl delete mode 100644 src/arrays/stiefel_lie_algebra_horizontal.jl delete mode 100644 src/arrays/stiefel_projection.jl delete mode 100644 src/arrays/symmetric.jl delete mode 100644 src/arrays/triangular.jl delete mode 100644 src/arrays/upper_triangular.jl delete mode 100644 src/manifolds/abstract_manifold.jl delete mode 100644 src/manifolds/grassmann_manifold.jl delete mode 100644 src/manifolds/stiefel_manifold.jl delete mode 100644 src/optimizers/go_bridges.jl diff --git a/src/GeometricMachineLearning.jl b/src/GeometricMachineLearning.jl index 42dc63eb4..445598608 100644 --- a/src/GeometricMachineLearning.jl +++ b/src/GeometricMachineLearning.jl @@ -24,16 +24,42 @@ import SymbolicNeuralNetworks: SymbolicPullback using SymbolicNeuralNetworks: derivative, SymbolicNeuralNetwork import Symbolics -# Only the names GML actually uses are brought in: GeometricOptimizers exports ~20 names that -# GML defines itself (`Manifold`, `StiefelManifold`, `SkewSymMatrix`, `Optimizer`, `rgrad`, …), -# and a blanket `using` makes redefining them an error on Julia 1.10. +# The manifolds, the structured matrix types, the global sections and the retractions are +# `GeometricOptimizers`' — GML used to carry near-verbatim copies of all eleven types, which Julia +# saw as *distinct* from the upstream ones, so none of GeometricOptimizers' generic machinery +# dispatched on them and GML re-implemented the retraction pipeline four times over. See +# [#234](https://github.com/JuliaGNI/GeometricMachineLearning.jl/issues/234). +# +# `import` and not `using ...: ...`: GML adds constructor methods to several of these types (in +# `layers/`, `arrays/gml_extensions.jl` and the kernels), and extending a *type* reached through +# `using` warns on every such method since Julia 1.12 — "Constructor for type … was extended in +# `GeometricMachineLearning` without explicit qualification or import". Everything imported here is +# re-exported below, which is what keeps `using GeometricMachineLearning` alone sufficient. import GeometricOptimizers -using GeometricOptimizers: OptimizerSolution, Cayley, Geodesic, cayley, geodesic, retraction, - apply_section, apply_section!, OptimizerMethod, - GradientCache, MomentumCache, AdamCache, - GradientMethod, MomentumMethod, Adam, - GradientState, MomentumState, AdamState, - GlobalSection, global_rep +import GeometricOptimizers: Manifold, StiefelManifold, GrassmannManifold +import GeometricOptimizers: SkewSymMatrix, SymmetricMatrix, AbstractTriangular, + LowerTriangular, UpperTriangular, StiefelProjection +import GeometricOptimizers: AbstractLieAlgHorMatrix, StiefelLieAlgHorMatrix, + GrassmannLieAlgHorMatrix +import GeometricOptimizers: rgrad, metric, check, Ω, global_section +# `assign_columns(Q, N, n)` — the first `n` columns of a `QR` factor, allocated on `Q`'s backend. +# It is upstream's, and internal there; the three manifold layers below initialise their weights +# with it. +import GeometricOptimizers: assign_columns +import GeometricOptimizers: GlobalSection, global_rep, apply_section, apply_section!, + update_section! +import GeometricOptimizers: AbstractRetraction, Geodesic, Cayley, geodesic, cayley, retraction +import GeometricOptimizers: OptimizerMethod, OptimizerSolution, + GradientMethod, MomentumMethod, Adam, + GradientState, MomentumState, AdamState, + AdamOptimizerWithDecay, DecayingStatic +import GeometricOptimizers: update! +# `solve!` is imported rather than started afresh so that GML's `solve!(::NeuralNetwork{<:PSDArch}, +# …)` — solve for the parameters directly, by SVD, instead of training for them — is a method of the +# same verb a caller already has from GeometricOptimizers, and not a second function of the name. +import GeometricOptimizers: solve! +# The optimizer *caches* stay internal upstream — they are `solver_step!` scratch — so GML reaches +# them as `GeometricOptimizers.AdamCache` where it needs to name one, and no longer re-exports them. import AbstractNeuralNetworks: Architecture, Model, AbstractExplicitLayer, AbstractExplicitCell, AbstractNeuralNetwork, NeuralNetwork, @@ -44,7 +70,11 @@ import AbstractNeuralNetworks: Chain, GridCell import AbstractNeuralNetworks: input_dimension, output_dimension import AbstractNeuralNetworks: Dense, Linear, Recurrent import AbstractNeuralNetworks: IdentityActivation, ZeroVector -import AbstractNeuralNetworks: add!, update! +# `update!` used to be imported here too, from `AbstractNeuralNetworks`, and re-exported. GML never +# added a method to it, so all the export did was shadow `GeometricOptimizers.update!` — which is +# `GeometricBase.update!`, a different generic function, and the one that actually has methods for +# the optimizer caches. It is imported from GeometricOptimizers with the rest of them below. +import AbstractNeuralNetworks: add! import AbstractNeuralNetworks: layer import AbstractNeuralNetworks: initialparameters import AbstractNeuralNetworks: parameterlength @@ -75,23 +105,22 @@ include("utils.jl") include("data_loader/data_loader.jl") -# INCLUDE ARRAYS -include("arrays/skew_symmetric.jl") -include("arrays/symmetric.jl") +# INCLUDE ARRAYS — the structured matrix types come from GeometricOptimizers; `PoissonTensor` is +# GML's own, and `gml_extensions.jl` holds what GML adds to the upstream types. include("arrays/poisson_tensor.jl") -include("arrays/abstract_lie_algebra_horizontal.jl") -include("arrays/stiefel_lie_algebra_horizontal.jl") -include("arrays/grassmann_lie_algebra_horizontal.jl") -include("arrays/triangular.jl") -include("arrays/lower_triangular.jl") -include("arrays/upper_triangular.jl") - -export SymmetricMatrix, PoissonTensor, SkewSymMatrix -export StiefelLieAlgHorMatrix -export SymplecticLieAlgMatrix, SymplecticLieAlgHorMatrix -export GrassmannLieAlgHorMatrix -export StiefelProjection, SymplecticProjection +include("arrays/gml_extensions.jl") + +# Re-exported from GeometricOptimizers, so that `using GeometricMachineLearning` on its own still +# gives a caller the matrix types its layers are parametrized by. +export SymmetricMatrix, SkewSymMatrix export LowerTriangular, UpperTriangular +export StiefelLieAlgHorMatrix, GrassmannLieAlgHorMatrix +export StiefelProjection +# GML's own +export PoissonTensor +# `SymplecticLieAlgMatrix`, `SymplecticLieAlgHorMatrix` and `SymplecticProjection` used to be +# exported here. Nothing has defined them for as long as the git history goes back, so the exports +# were silent `UndefVarError`s waiting for a caller; see `test/exports.jl`. include("kernels/assign_q_and_p.jl") include("kernels/tensor_mat_mul.jl") @@ -143,15 +172,9 @@ export LinearSymplecticLayerP, LinearSymplecticLayerQ # `SymplecticStiefelLayer` used to be exported here; the file defining it # (`layers/symplectic_stiefel_layer.jl`) is commented out below, so the name never existed. -include("manifolds/abstract_manifold.jl") -include("manifolds/stiefel_manifold.jl") -# include("manifolds/symplectic_stiefel_manifold.jl") -include("manifolds/grassmann_manifold.jl") - -include("arrays/stiefel_projection.jl") - +# The manifolds are GeometricOptimizers' too, along with the geometry that goes with them. export StiefelManifold, GrassmannManifold, Manifold -export rgrad, metric +export rgrad, metric, check include("layers/sympnets.jl") include("layers/bias_layer.jl") @@ -180,29 +203,35 @@ export ResNet export Transformer export TransformerIntegrator, StandardTransformerIntegrator -# INCLUDE OPTIMIZERS — types come from GeometricOptimizers -include("optimizers/go_bridges.jl") +# INCLUDE OPTIMIZERS — the methods, states, sections and retractions come from GeometricOptimizers. +# `go_bridges.jl` used to sit here: thirty-odd methods reconnecting GML's copies of the structured +# types to GeometricOptimizers' `_add!`/`_rac!`/`_square!`/`_div!`/`_rmul!`/`update_section!`. The +# types are the same objects now, so upstream's own methods apply and the file is gone. include("optimizers/optimizer.jl") -export OptimizerMethod, AbstractCache -export GradientMethod, GradientCache, GradientState -export MomentumMethod, MomentumCache, MomentumState -export Adam, AdamCache, AdamState +export OptimizerMethod +export GradientMethod, GradientState +export MomentumMethod, MomentumState +export Adam, AdamState export Optimizer export optimization_step! -export GlobalSection, apply_section, apply_section! +export GlobalSection, global_section, apply_section, apply_section!, update_section! export global_rep export Geodesic, Cayley export geodesic, cayley export retraction export update! -export check +# `AbstractCache` and the three cache types used to be exported here. The caches are +# `solver_step!` scratch and stay internal to GeometricOptimizers, for every method alike; reach one +# as `GeometricOptimizers.AdamCache` if you genuinely need to name it. # backward-compat aliases (old names → new names) const GradientOptimizer = GradientMethod const MomentumOptimizer = MomentumMethod const AdamOptimizer = Adam export GradientOptimizer, MomentumOptimizer, AdamOptimizer -export AdamOptimizerWithDecay +# Re-exported from GeometricOptimizers, which owns the one definition of them now. GML's own +# `AdamOptimizerWithDecay` was a second, incompatible export of the same name — issue B1. +export AdamOptimizerWithDecay, DecayingStatic #INCLUDE ABSTRACT TRAINING integrator export AbstractTrainingMethod diff --git a/src/architectures/transformer_neural_network.jl b/src/architectures/transformer_neural_network.jl index 3cbca975b..4076cff5c 100644 --- a/src/architectures/transformer_neural_network.jl +++ b/src/architectures/transformer_neural_network.jl @@ -11,7 +11,7 @@ The optional keyword arguments are: - `n_heads::Int=7`: The number of heads in the `MultiHeadAttention` (mha) layers. - `L::Int=16`: The number of transformer blocks. - `activation=softmax`: The activation function. -- `Stiefel::Bool=true`: Whether the matrices in the mha layers are on the [`StiefelManifold`](@ref). +- `Stiefel::Bool=true`: Whether the matrices in the mha layers are on the [`StiefelManifold`](@extref GeometricOptimizers GeometricOptimizers.StiefelManifold). - `add_connection::Bool=true`: Whether the input is appended to the output of the mha layer. (skip connection) """ struct ClassificationTransformer{AT} <: Architecture diff --git a/src/arrays/abstract_lie_algebra_horizontal.jl b/src/arrays/abstract_lie_algebra_horizontal.jl deleted file mode 100644 index 72a9a30c9..000000000 --- a/src/arrays/abstract_lie_algebra_horizontal.jl +++ /dev/null @@ -1,8 +0,0 @@ -@doc raw""" - AbstractLieAlgHorMatrix <: AbstractMatrix - -`AbstractLieAlgHorMatrix` is a supertype for various horizontal components of Lie algebras. We usually call this ``\mathfrak{g}^\mathrm{hor}``. - -See [`StiefelLieAlgHorMatrix`](@ref) and [`GrassmannLieAlgHorMatrix`](@ref) for concrete examples. -""" -abstract type AbstractLieAlgHorMatrix{T} <: AbstractMatrix{T} end diff --git a/src/arrays/gml_extensions.jl b/src/arrays/gml_extensions.jl new file mode 100644 index 000000000..85de234ed --- /dev/null +++ b/src/arrays/gml_extensions.jl @@ -0,0 +1,43 @@ +# What `GeometricMachineLearning` adds to `GeometricOptimizers`' structured matrix types. +# +# The types themselves — `SkewSymMatrix`, `SymmetricMatrix`, the triangular family, the +# Lie-algebra-horizontal lifts, `StiefelProjection` and the manifolds — are imported from +# `GeometricOptimizers` rather than redefined here; see the import block in +# `GeometricMachineLearning.jl`. This file holds the methods that are genuinely GML's, i.e. the ones +# that reach for a dependency `GeometricOptimizers` does not have. +# +# There is only one such family. `add!` is `AbstractNeuralNetworks.add!`, a *different generic +# function* from the `add!` `GeometricOptimizers` defines internally for the same types, so the +# upstream methods do not serve GML's callers. It is exported by GML and has been since 0.1. +# +# `networkbackend` needs no methods at all: `AbstractNeuralNetworks.networkbackend(::AbstractArray)` +# forwards to `KernelAbstractions.get_backend`, and `GeometricOptimizers` implements that for every +# one of these types. GML used to define both halves. + +function add!(C::SkewSymMatrix, A::SkewSymMatrix, B::SkewSymMatrix) + @assert A.n == B.n == C.n + add!(C.S, A.S, B.S) +end + +function add!(C::SymmetricMatrix, A::SymmetricMatrix, B::SymmetricMatrix) + @assert A.n == B.n == C.n + add!(C.S, A.S, B.S) +end + +function add!(C::AT, A::AT, B::AT) where {AT <: AbstractTriangular} + @assert A.n == B.n == C.n + add!(C.S, A.S, B.S) +end + +function add!(C::StiefelLieAlgHorMatrix, A::StiefelLieAlgHorMatrix, B::StiefelLieAlgHorMatrix) + @assert A.N == B.N == C.N + @assert A.n == B.n == C.n + add!(C.A, A.A, B.A) + add!(C.B, A.B, B.B) +end + +function add!(C::GrassmannLieAlgHorMatrix, A::GrassmannLieAlgHorMatrix, B::GrassmannLieAlgHorMatrix) + @assert A.N == B.N == C.N + @assert A.n == B.n == C.n + add!(C.B, A.B, B.B) +end diff --git a/src/arrays/grassmann_lie_algebra_horizontal.jl b/src/arrays/grassmann_lie_algebra_horizontal.jl deleted file mode 100644 index a2a450a87..000000000 --- a/src/arrays/grassmann_lie_algebra_horizontal.jl +++ /dev/null @@ -1,209 +0,0 @@ -@doc raw""" - GrassmannLieAlgHorMatrix(B::AbstractMatrix, N::Integer, n::Integer) - -Build an instance of `GrassmannLieAlgHorMatrix` based on an arbitrary matrix `B` of size ``(N-n)\times{}n``. - -`GrassmannLieAlgHorMatrix` is the *horizontal component of the Lie algebra of skew-symmetric matrices* (with respect to the canonical metric). - -# Extended help - -The projection here is: ``\pi:S \to SE/\sim`` where -```math -E = \begin{bmatrix} \mathbb{I}_{n} \\ \mathbb{O}_{(N-n)\times{}n} \end{bmatrix}, -``` - -and the equivalence relation is - -```math -V_1 \sim V_2 \iff \exists A\in\mathcal{S}_\mathrm{skew}(n) \text{ such that } V_2 = V_1 + \begin{bmatrix} A \\ \mathbb{O} \end{bmatrix} -``` - -An element of GrassmannLieAlgMatrix takes the form: -```math -\begin{pmatrix} -\bar{\mathbb{O}} & B^T \\ B & \mathbb{O} -\end{pmatrix}, -``` -where ``\bar{\mathbb{O}}\in\mathbb{R}^{n\times{}n}`` and ``\mathbb{O}\in\mathbb{R}^{(N - n)\times(N-n)}.`` -""" -mutable struct GrassmannLieAlgHorMatrix{T, ST <: AbstractMatrix{T}} <: AbstractLieAlgHorMatrix{T} - B::ST - N::Int - n::Int - - #maybe modify this - you don't need N & n as inputs! - function GrassmannLieAlgHorMatrix(B::AbstractMatrix{T}, N::Int, n::Int) where {T} - @assert n == size(B,2) - @assert N == size(B,1) + n - - new{T, typeof(B)}(B, N, n) - end -end - -@doc raw""" - GrassmannLieAlgHorMatrix(D::AbstractMatrix, n::Integer) - -Take a big matrix as input and build an instance of `GrassmannLieAlgHorMatrix`. - -The integer ``N`` in ``Gr(n, N)`` here is the number of rows of `D`. - -# Extended help - -If the constructor is called with a big ``N\times{}N`` matrix, then the projection is performed the following way: - -```math -\begin{pmatrix} -A & B_1 \\ -B_2 & D -\end{pmatrix} \mapsto -\begin{pmatrix} -\bar{\mathbb{O}} & -B_2^T \\ -B_2 & \mathbb{O} -\end{pmatrix}. -``` - -This can also be seen as the operation: -```math -D \mapsto \Omega(E, DE - EE^TDE), -``` - -where ``\Omega`` is the horizontal lift [`GeometricMachineLearning.Ω`](@ref). -""" -function GrassmannLieAlgHorMatrix(D::AbstractMatrix, n::Int) - N = size(D, 1) - @assert N ≥ n - - @views B = D[(n + 1):N,1:n] - GrassmannLieAlgHorMatrix(B, N, n) -end - -Base.parent(A::GrassmannLieAlgHorMatrix) = (A.B, ) -Base.size(A::GrassmannLieAlgHorMatrix) = (A.N, A.N) - -networkbackend(B::GrassmannLieAlgHorMatrix) = networkbackend(B.B) - -function Base.getindex(A::GrassmannLieAlgHorMatrix{T}, i::Integer, j::Integer) where {T} - if i ≤ A.n - if j ≤ A.n - return T(0.) - end - return -A.B[j - A.n, i] - end - if j ≤ A.n - return A.B[i - A.n, j] - end - return T(0.) -end - -function Base.:+(A::GrassmannLieAlgHorMatrix, B::GrassmannLieAlgHorMatrix) - @assert A.N == B.N - @assert A.n == B.n - GrassmannLieAlgHorMatrix(A.B + B.B, - A.N, - A.n) -end - -function Base.:-(A::GrassmannLieAlgHorMatrix, B::GrassmannLieAlgHorMatrix) - @assert A.N == B.N - @assert A.n == B.n - GrassmannLieAlgHorMatrix(A.B - B.B, - A.N, - A.n) -end - -function add!(C::GrassmannLieAlgHorMatrix, A::GrassmannLieAlgHorMatrix, B::GrassmannLieAlgHorMatrix) - @assert A.N == B.N == C.N - @assert A.n == B.n == C.n - add!(C.B, A.B, B.B) -end - -function Base.:-(A::GrassmannLieAlgHorMatrix) - GrassmannLieAlgHorMatrix( -A.B, A.N, A.n) -end - -function Base.:*(A::GrassmannLieAlgHorMatrix, α::Real) - GrassmannLieAlgHorMatrix( α*A.B, A.N, A.n) -end - -Base.:*(α::Real, A::GrassmannLieAlgHorMatrix) = A*α - -function Base.zeros(::Type{GrassmannLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - GrassmannLieAlgHorMatrix( - zeros(T, N-n, n), - N, - n - ) -end - -function Base.zeros(::Type{GrassmannLieAlgHorMatrix}, N::Integer, n::Integer) - GrassmannLieAlgHorMatrix( - zeros(N-n, n), - N, - n - ) -end - -function Base.zeros(backend::KernelAbstractions.Backend, ::Type{GrassmannLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - GrassmannLieAlgHorMatrix( - KernelAbstractions.zeros(backend, T, N-n, n), - N, - n - ) -end - -Base.similar(A::GrassmannLieAlgHorMatrix, dims::Union{Integer, AbstractUnitRange}...) = zeros(GrassmannLieAlgHorMatrix{eltype(A)}, dims...) -Base.similar(A::GrassmannLieAlgHorMatrix) = zeros(GrassmannLieAlgHorMatrix{eltype(A)}, A.N, A.n) -Base.zero(A::GrassmannLieAlgHorMatrix) = zeros(GrassmannLieAlgHorMatrix{eltype(A)}, A.N, A.n) - -function Base.rand(rng::Random.AbstractRNG, ::Type{GrassmannLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - GrassmannLieAlgHorMatrix(rand(rng, T, N-n, n), N, n) -end - -function Base.rand(rng::Random.AbstractRNG, ::Type{GrassmannLieAlgHorMatrix}, N::Integer, n::Integer) - GrassmannLieAlgHorMatrix(rand(rng, N-n, n), N, n) -end - -function Base.rand(::Type{GrassmannLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - rand(Random.default_rng(), GrassmannLieAlgHorMatrix{T}, N, n) -end - -function Base.rand(::Type{GrassmannLieAlgHorMatrix}, N::Integer, n::Integer) - rand(Random.default_rng(), GrassmannLieAlgHorMatrix, N, n) -end - -function scalar_add(A::GrassmannLieAlgHorMatrix, δ::Real) - GrassmannLieAlgHorMatrix(A.B .+ δ, A.N, A.n) -end - -#define these functions more generally! (maybe make a fallback script!!) -function ⊙²(A::GrassmannLieAlgHorMatrix) - GrassmannLieAlgHorMatrix(A.B.^2, A.N, A.n) -end -function racᵉˡᵉ(A::GrassmannLieAlgHorMatrix) - GrassmannLieAlgHorMatrix(sqrt.(A.B), A.N, A.n) -end -function /ᵉˡᵉ(A::GrassmannLieAlgHorMatrix, B::GrassmannLieAlgHorMatrix) - GrassmannLieAlgHorMatrix(A.B./B.B, A.N, A.n) -end - -function LinearAlgebra.mul!(C::GrassmannLieAlgHorMatrix, A::GrassmannLieAlgHorMatrix, α::Real) - mul!(C.B, A.B, α) -end -LinearAlgebra.mul!(C::GrassmannLieAlgHorMatrix, α::Real, A::GrassmannLieAlgHorMatrix) = mul!(C, A, α) -LinearAlgebra.rmul!(C::GrassmannLieAlgHorMatrix, α::Real) = mul!(C, C, α) - -function _round(B::GrassmannLieAlgHorMatrix; kwargs...) - GrassmannLieAlgHorMatrix( - _round(B.B; kwargs...), - B.N, - B.n - ) -end - -function Base.copyto!(A::GrassmannLieAlgHorMatrix, B::GrassmannLieAlgHorMatrix) - copyto!(A.B, B.B) - A -end - -# fills the *storage*; see the comment on `fill!(::SkewSymMatrix, ::Any)` -Base.fill!(A::GrassmannLieAlgHorMatrix, val) = (fill!(A.B, val); A) diff --git a/src/arrays/lower_triangular.jl b/src/arrays/lower_triangular.jl deleted file mode 100644 index 7f1e3bb9e..000000000 --- a/src/arrays/lower_triangular.jl +++ /dev/null @@ -1,96 +0,0 @@ -@doc raw""" - LowerTriangular(S::AbstractVector, n::Int) - -Build a lower-triangular matrix from a vector. - -A lower-triangular matrix is an ``n\times{}n`` matrix that has zeros on the diagonal and on the upper triangular. - -The data are stored in a vector ``S`` similarly to other matrices. See [`UpperTriangular`](@ref), [`SkewSymMatrix`](@ref) and [`SymmetricMatrix`](@ref). - -The struct two fields: `S` and `n`. The first stores all the entries of the matrix in a sparse fashion (in a vector) and the second is the dimension ``n`` for ``A\in\mathbb{R}^{n\times{}n}``. - -# Examples -```jldoctest -using GeometricMachineLearning -S = [1, 2, 3, 4, 5, 6] -LowerTriangular(S, 4) - -# output - -4×4 LowerTriangular{Int64, Vector{Int64}}: - 0 0 0 0 - 1 0 0 0 - 2 3 0 0 - 4 5 6 0 -``` -""" -mutable struct LowerTriangular{T, AT <: AbstractVector{T}} <: AbstractTriangular{T} - S::AT - n::Int -end - -@doc raw""" - LowerTriangular(A::AbstractMatrix) - -Build a lower-triangular matrix from a matrix. - -This is done by taking the lower left of that matrix. - -# Examples -```jldoctest -using GeometricMachineLearning -M = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16] -LowerTriangular(M) - -# output - -4×4 LowerTriangular{Int64, Vector{Int64}}: - 0 0 0 0 - 5 0 0 0 - 9 10 0 0 - 13 14 15 0 -``` -""" -function LowerTriangular(S::AbstractMatrix{T}) where {T} - n = size(S, 1) - @assert size(S, 2) == n - S_vec = map_to_lo(S) - LowerTriangular(S_vec, n) -end - -function Base.getindex(A::LowerTriangular{T}, i::Int, j::Int) where T - if j == i - return zero(T) - end - if i > j - return A.S[(i-2) * (i-1) ÷ 2 + j] - end - return zero(T) -end - -@kernel function lo_mat_mul_kernel!(C::AbstractMatrix{T}, S::AbstractVector{T}, B::AbstractMatrix{T}, n) where T - i, j = @index(Global, NTuple) - - tmp_sum = zero(T) - for k = 1:(i-1) - tmp_sum += S[(i-2)*(i-1)÷2+k] * B[k, j] - end - C[i,j] = tmp_sum -end - -function map_to_lo(A::AbstractMatrix{T}) where T - n = size(A, 1) - @assert size(A, 2) == n - backend = networkbackend(A) - S = KernelAbstractions.zeros(backend, T, n * (n - 1) ÷ 2) - assign_Skew_val! = assign_Skew_val_kernel!(backend) - for i in 2:n - assign_Skew_val!(S, A, i, ndrange = (i - 1)) - end - S -end - -# define routines for generalizing ChainRulesCore to LowerTriangular -ChainRulesCore.ProjectTo(A::AT) where AT <: LowerTriangular = ProjectTo{AT}(; triang = ProjectTo(A.S)) -(project::ProjectTo{<:LowerTriangular})(dA::AbstractMatrix) = LowerTriangular(project.triang(map_to_lo(dA)), size(dA, 2)) -(project::ProjectTo{<:LowerTriangular})(dA::LowerTriangular) = LowerTriangular(project.triang(dA.S), dA.n) \ No newline at end of file diff --git a/src/arrays/skew_symmetric.jl b/src/arrays/skew_symmetric.jl deleted file mode 100644 index 362c6b07a..000000000 --- a/src/arrays/skew_symmetric.jl +++ /dev/null @@ -1,357 +0,0 @@ -@doc raw""" - SkewSymMatrix(S::AbstractVector, n::Integer) - -Instantiate a skew-symmetric matrix with information stored in vector `S`. - -A skew-symmetric matrix ``A`` is a matrix ``A^T = -A``. - -Internally the `struct` saves a vector ``S`` of size ``n(n-1)\div2``. The conversion is done the following way: -```math -[A]_{ij} = \begin{cases} 0 & \text{if $i=j$} \\ - S[( (i-2) (i-1) ) \div 2 + j] & \text{if $i>j$}\\ - S[( (j-2) (j-1) ) \div 2 + i] & \text{else}. \end{cases} -``` - -So ``S`` stores a string of vectors taken from ``A``: ``S = [\tilde{a}_1, \tilde{a}_2, \ldots, \tilde{a}_n]`` with ``\tilde{a}_i = [[A]_{i1},[A]_{i2},\ldots,[A]_{i(i-1)}]``. - -Also see [`SymmetricMatrix`](@ref), [`LowerTriangular`](@ref) and [`UpperTriangular`](@ref). - -# Examples -```jldoctest -using GeometricMachineLearning -S = [1, 2, 3, 4, 5, 6] -SkewSymMatrix(S, 4) - -# output - -4×4 SkewSymMatrix{Int64, Vector{Int64}}: - 0 -1 -2 -4 - 1 0 -3 -5 - 2 3 0 -6 - 4 5 6 0 -``` -""" -mutable struct SkewSymMatrix{T, AT <: AbstractVector{T}} <: AbstractMatrix{T} - S::AT - n::Int - - function SkewSymMatrix(S::AbstractVector{T},n::Int) where {T} - @assert length(S) == n*(n-1)÷2 - new{T,typeof(S)}(S,n) - end -end - -@doc raw""" - SkewSymMatrix(A::AbstractMatrix) - -Perform `0.5 * (A - A')` and store the matrix in an efficient way (as a vector with ``n(n-1)/2`` entries). - -If the constructor is called with a matrix as input it returns a skew-symmetric matrix via the projection: -```math -A \mapsto \frac{1}{2}(A - A^T). -``` - -# Examples -```jldoctest -using GeometricMachineLearning -M = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16] -SkewSymMatrix(M) - -# output - -4×4 SkewSymMatrix{Float64, Vector{Float64}}: - 0.0 -1.5 -3.0 -4.5 - 1.5 0.0 -1.5 -3.0 - 3.0 1.5 0.0 -1.5 - 4.5 3.0 1.5 0.0 -``` - -# Extended help - -Note that the constructor is designed in such a way that it always returns matrices of type `SkewSymMatrix{<:AbstractFloat}` when called with a matrix, even if this matrix is of type `AbstractMatrix{<:Integer}`. - -If the user wishes to allocate a matrix `SkewSymMatrix{<:Integer}` then call: - -```julia -SkewSymMatrix(::AbstractVector, n::Integer) -``` - -Note that this is different from [`LowerTriangular`](@ref) and [`UpperTriangular`](@ref) as no porjection takes place there. -""" -function SkewSymMatrix(S::AbstractMatrix{T}) where {T} - n = size(S, 1) - @assert size(S, 2) == n - S_vec = map_to_Skew(S) - SkewSymMatrix(S_vec, n) -end - -function return_element(S::AbstractVector{T}, i::Int, j::Int) where T - if j == i - return zero(T) - end - if i > j - return S[(i-2) * (i-1) ÷ 2 + j] - end - return - S[ (j-2) * (j-1) ÷ 2 + i] -end - -function Base.getindex(A::SkewSymMatrix, i::Int, j::Int) - return_element(A.S, i, j) -end - - -Base.parent(A::SkewSymMatrix) = A.S -Base.size(A::SkewSymMatrix) = (A.n,A.n) - -@kernel function addition_kernel!(C::AbstractMatrix, S::AbstractVector, B::AbstractMatrix) - i, j = @index(Global, NTuple) - C[i, j] = return_element(S, i, j) + B[i, j] -end - -function Base.:+(A::SkewSymMatrix{T}, B::AbstractMatrix{T}) where T - @assert size(A) == size(B) - backend = networkbackend(B) - addition! = addition_kernel!(backend) - C = KernelAbstractions.allocate(backend, T, size(A)...) - addition!(C, A.S, B; ndrange = size(A)) - - C -end - -Base.:+(B::AbstractMatrix, A::SkewSymMatrix) = B + A - -function Base.:+(A::SkewSymMatrix, B::SkewSymMatrix) - @assert A.n == B.n - SkewSymMatrix(A.S + B.S, A.n) -end - -function add!(C::SkewSymMatrix, A::SkewSymMatrix, B::SkewSymMatrix) - @assert A.n == B.n == C.n - add!(C.S, A.S, B.S) -end - -function Base.:-(A::SkewSymMatrix, B::SkewSymMatrix) - @assert A.n == B.n - SkewSymMatrix(A.S - B.S, A.n) -end - -function Base.:-(A::SkewSymMatrix) - SkewSymMatrix(-A.S, A.n) -end - -function Base.:*(A::SkewSymMatrix, α::Real) - SkewSymMatrix(α*A.S, A.n) -end - -Base.:*(α::Real, A::SkewSymMatrix) = A*α - -function Base.zeros(::Type{SkewSymMatrix{T}}, n::Int) where T - zeros(CPU(), SkewSymMatrix{T}, n) -end - -function Base.zeros(backend::KernelAbstractions.Backend, ::Type{SkewSymMatrix{T}}, n::Int) where T - zero_vec = if n != 1 - KernelAbstractions.zeros(backend, T, n*(n-1)÷2) - else - KernelAbstractions.allocate(backend, T, n*(n-1)÷2) - end - SkewSymMatrix(zero_vec, n) -end - -function Base.zeros(::Type{SkewSymMatrix}, n::Int) - SkewSymMatrix(zeros(n*(n-1)÷2), n) -end - -function Base.rand(rng::Random.AbstractRNG, ::Type{SkewSymMatrix{T}}, n::Int) where T - SkewSymMatrix(rand(rng, T, n*(n-1)÷2),n) -end - -function Base.rand(rng::Random.AbstractRNG, ::Type{SkewSymMatrix}, n::Int) - SkewSymMatrix(rand(rng, n*(n-1)÷2), n) -end - -# TODO: make defaults when no rng is specified!!! (prbabaly rng ← Random.default_rng()) -function Base.rand(type::Type{SkewSymMatrix{T}}, n::Integer) where T - rand(Random.default_rng(), type, n) -end - -function Base.rand(type::Type{SkewSymMatrix}, n::Integer) - rand(Random.default_rng(), type, n) -end - -function Base.rand(rng::AbstractRNG, backend::KernelAbstractions.Backend, type::Type{SkewSymMatrix{T}}, n::Integer) where T - S = KernelAbstractions.allocate(backend, T, n*(n-1)÷2) - Random.rand!(rng, S) - SkewSymMatrix(S, n) -end - -function Base.rand(backend::KernelAbstractions.Backend, type::Type{SkewSymMatrix{T}}, n::Integer) where T - rand(Random.default_rng(), backend, type, n) -end - -#these are Adam operations: -function scalar_add(A::SkewSymMatrix, δ::Real) - SkewSymMatrix(A.S .+ δ, A.n) -end - -#element-wise squares and square root (for Adam) -function ⊙²(A::SkewSymMatrix) - SkewSymMatrix(A.S.^2, A.n) -end -function racᵉˡᵉ(A::SkewSymMatrix) - SkewSymMatrix(sqrt.(A.S), A.n) -end -function /ᵉˡᵉ(A::SkewSymMatrix, B::SkewSymMatrix) - @assert A.n == B.n - SkewSymMatrix(A.S ./ B.S, A.n) -end - -function LinearAlgebra.mul!(C::SkewSymMatrix, A::SkewSymMatrix, α::Real) - mul!(C.S, A.S, α) -end -LinearAlgebra.mul!(C::SkewSymMatrix, α::Real, A::SkewSymMatrix) = mul!(C, A, α) -LinearAlgebra.rmul!(C::SkewSymMatrix, α::Real) = mul!(C, C, α) - -function Base.:*(A::SkewSymMatrix{T}, B::AbstractMatrix{T}) where T - m1, m2 = size(B) - @assert m1 == A.n - backend = networkbackend(A) - C = KernelAbstractions.allocate(backend, T, A.n, m2) - - skew_mat_mul! = skew_mat_mul_kernel!(backend) - skew_mat_mul!(C, A.S, B, A.n, ndrange=size(C)) - C -end - -@kernel function skew_mat_mul_kernel!(C::AbstractMatrix{T}, S::AbstractVector{T}, B::AbstractMatrix{T}, n) where T - i, j = @index(Global, NTuple) - - tmp_sum = zero(T) - for k = 1:(i-1) - tmp_sum += S[(i-2)*(i-1)÷2+k] * B[k, j] - end - for k = (i+1):n - tmp_sum += -S[(k-2)*(k-1)÷2+i] * B[k, j] - end - C[i,j] = tmp_sum -end - -function Base.:*(B::AbstractMatrix{T}, A::SkewSymMatrix{T}) where T - (-A*B')' -end - -function Base.:*(A::SkewSymMatrix, b::AbstractVector{T}) where T - A*reshape(b, length(b), 1) -end - -function Base.one(A::SkewSymMatrix{T}) where T - backend = networkbackend(A.S) - unit_matrix = KernelAbstractions.zeros(backend, T, A.n, A.n) - write_ones! = write_ones_kernel!(backend) - write_ones!(unit_matrix, ndrange=A.n) - unit_matrix -end - - -# the first matrix is multiplied onto A2 in order for it to not be SkewSymMatrix! -function Base.:*(A1::SkewSymMatrix{T}, A2::SkewSymMatrix{T}) where T - A1 * (one(A2) * A2) -end - -@doc raw""" - vec(A) - -Output the associated vector of `A`. - -# Examples - -```jldoctest -using GeometricMachineLearning - -M = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16] -SkewSymMatrix(M) |> vec - -# output - -6-element Vector{Float64}: - 1.5 - 3.0 - 1.5 - 4.5 - 3.0 - 1.5 -``` -""" -function Base.vec(A::SkewSymMatrix) - A.S -end - -function Base.zero(A::SkewSymMatrix) - SkewSymMatrix(zero(A.S), A.n) -end - -# see the comment on `similar(::SymmetricMatrix)` -Base.similar(A::SkewSymMatrix) = SkewSymMatrix(similar(A.S), A.n) - -function networkbackend(A::SkewSymMatrix) - networkbackend(A.S) -end - -function assign!(B::SkewSymMatrix{T}, C::SkewSymMatrix{T}) where T - B.S .= C.S -end - -function Base.copy(A::SkewSymMatrix) - SkewSymMatrix(copy(A.S), A.n) -end - -@kernel function assign_Skew_val_kernel!(S, A_skew, i) - j = @index(Global) - S[((i - 2) * (i - 1) ÷ 2 + j)] = A_skew[i, j] -end - -function map_to_Skew(A::AbstractMatrix{T}) where T - n = size(A, 1) - @assert size(A, 2) == n - A_skew = T(.5)*(A - A') - backend = networkbackend(A) - S = if n != 1 - KernelAbstractions.zeros(backend, T, n * (n - 1) ÷ 2) - else - KernelAbstractions.allocate(backend, T, n * (n - 1) ÷ 2) - end - assign_Skew_val! = assign_Skew_val_kernel!(backend) - for i in 2:n - assign_Skew_val!(S, A_skew, i, ndrange = (i - 1)) - end - S -end - -function map_to_Skew(A::AbstractMatrix{T}) where T <: Integer - Float = T == Int64 ? Float64 : Float32 - map_to_Skew(Float.(A)) -end - -function Base.copyto!(A::SkewSymMatrix, B::SkewSymMatrix) - A.S .= B.S - nothing -end - -function _round(A::SkewSymMatrix; kwargs...) - SkewSymMatrix(_round(A.S; kwargs...), A.n) -end - -function _round(A::AbstractArray; kwargs...) - round.(A; kwargs...) -end - -# this fills the *storage*: `fill!(A, val)` gives a matrix whose strict lower triangle is `val`, whose -# strict upper triangle is `-val` and whose diagonal stays zero. A skew-symmetric matrix cannot hold a -# constant, and this is the only sensible reading of `fill!` for it. The optimizer caches use it to -# poison scratch arrays with `NaN`, where the sign does not matter. -Base.fill!(A::SkewSymMatrix, val) = (fill!(A.S, val); A) - -# define routines for generalizing ChainRulesCore to SkewSymMatrix -ChainRulesCore.ProjectTo(A::SkewSymMatrix) = ProjectTo{SkewSymMatrix}(; skew_sym = ProjectTo(A.S)) -(project::ProjectTo{SkewSymMatrix})(dA::AbstractMatrix) = SkewSymMatrix(project.skew_sym(map_to_Skew(dA)), size(dA, 2)) -(project::ProjectTo{SkewSymMatrix})(dA::SkewSymMatrix) = SkewSymMatrix(project.skew_sym(dA.S), dA.n) \ No newline at end of file diff --git a/src/arrays/stiefel_lie_algebra_horizontal.jl b/src/arrays/stiefel_lie_algebra_horizontal.jl deleted file mode 100644 index b61c47c52..000000000 --- a/src/arrays/stiefel_lie_algebra_horizontal.jl +++ /dev/null @@ -1,329 +0,0 @@ -@doc raw""" - StiefelLieAlgHorMatrix(A::SkewSymMatrix, B::AbstractMatrix, N::Integer, n::Integer) - -Build an instance of `StiefelLieAlgHorMatrix` based on a skew-symmetric matrix `A` and an arbitrary matrix `B`. - -An element of StiefelLieAlgMatrix takes the form: -```math -\begin{pmatrix} -A & B^T \\ B & \mathbb{O} -\end{pmatrix}, -``` -where ``A`` is skew-symmetric (this is [`SkewSymMatrix`](@ref) in `GeometricMachineLearning`). - -Also see [`GrassmannLieAlgHorMatrix`](@ref). - -# Extended help - -`StiefelLieAlgHorMatrix` is the *horizontal component of the Lie algebra of skew-symmetric matrices* (with respect to the canonical metric). - -The projection here is: ``\pi:S \to SE`` where -```math -E = \begin{bmatrix} \mathbb{I}_{n} \\ \mathbb{O}_{(N-n)\times{}n} \end{bmatrix}. -``` -The matrix ``E`` is implemented under [`StiefelProjection`](@ref) in `GeometricMachineLearning`. -""" -mutable struct StiefelLieAlgHorMatrix{T, AT <: SkewSymMatrix{T}, ST <: AbstractMatrix{T}} <: AbstractLieAlgHorMatrix{T} - A::AT - B::ST - N::Int - n::Int - - #maybe modify this - you don't need N & n as inputs! - function StiefelLieAlgHorMatrix(A::SkewSymMatrix{T}, B::AbstractMatrix{T}, N::Integer, n::Integer) where {T} - @assert n == A.n == size(B,2) - @assert N == size(B,1) + n - - new{T, typeof(A), typeof(B)}(A, B, N, n) - end -end - -@doc raw""" - StiefelLieAlgHorMatrix(D::AbstractMatrix, n::Integer) - -Take a big matrix as input and build an instance of `StiefelLieAlgHorMatrix`. - -The integer ``N`` in ``St(n, N)`` is the number of rows of `D`. - -# Extended help - -If the constructor is called with a big ``N\times{}N`` matrix, then the projection is performed the following way: - -```math -\begin{pmatrix} -A & B_1 \\ -B_2 & D -\end{pmatrix} \mapsto -\begin{pmatrix} -\mathrm{skew}(A) & -B_2^T \\ -B_2 & \mathbb{O} -\end{pmatrix}. -``` - -The operation ``\mathrm{skew}:\mathbb{R}^{n\times{}n}\to\mathcal{S}_\mathrm{skew}(n)`` is the skew-symmetrization operation. This is equivalent to calling of [`SkewSymMatrix`](@ref) with an ``n\times{}n`` matrix. - -This can also be seen as the operation: -```math -D \mapsto \Omega(E, DE) = \mathrm{skew}\left(2 \left(\mathbb{I} - \frac{1}{2} E E^T \right) DE E^T\right). -``` - -Also see [`GeometricMachineLearning.Ω`](@ref). -""" -function StiefelLieAlgHorMatrix(D::AbstractMatrix, n::Integer) - N = size(D, 1) - @assert N ≥ n - - @views A_small = SkewSymMatrix(D[1:n,1:n]) - @views B = D[(n + 1):N, 1:n] - StiefelLieAlgHorMatrix(A_small, B, N, n) -end - -Base.parent(A::StiefelLieAlgHorMatrix) = (A, B) -Base.size(A::StiefelLieAlgHorMatrix) = (A.N, A.N) - -function Base.getindex(A::StiefelLieAlgHorMatrix{T}, i, j) where {T} - if i ≤ A.n - if j ≤ A.n - return A.A[i, j] - end - return -A.B[j - A.n, i] - end - if j ≤ A.n - return A.B[i - A.n, j] - end - return T(0.) -end - -function Base.:+(A::StiefelLieAlgHorMatrix, B::StiefelLieAlgHorMatrix) - @assert A.N == B.N - @assert A.n == B.n - StiefelLieAlgHorMatrix( A.A + B.A, - A.B + B.B, - A.N, - A.n) -end - -function Base.:-(A::StiefelLieAlgHorMatrix, B::StiefelLieAlgHorMatrix) - @assert A.N == B.N - @assert A.n == B.n - StiefelLieAlgHorMatrix( A.A - B.A, - A.B - B.B, - A.N, - A.n) -end - -function add!(C::StiefelLieAlgHorMatrix, A::StiefelLieAlgHorMatrix, B::StiefelLieAlgHorMatrix) - @assert A.N == B.N == C.N - @assert A.n == B.n == C.n - add!(C.A, A.A, B.A) - add!(C.B, A.B, B.B) -end - - -function Base.:-(A::StiefelLieAlgHorMatrix) - StiefelLieAlgHorMatrix(-A.A, -A.B, A.N, A.n) -end - -function Base.:*(A::StiefelLieAlgHorMatrix, α::Real) - StiefelLieAlgHorMatrix( α*A.A, α*A.B, A.N, A.n) -end - -function Base.:+(B::StiefelLieAlgHorMatrix, A::AbstractMatrix) - @assert size(A) == size(B) - - C = copy(A) - @views C[1:B.n, 1:B.n] .= B.A + A[1:B.n, 1:B.n] - @views C[(B.n+1):B.N, 1:B.n] .= B.B + A[(B.n+1):B.N, 1:B.n] - @views C[1:B.n, (B.n+1):B.N] .= A[1:B.n, (B.n+1):B.N] - B.B' - - C -end - -Base.:+(A::AbstractMatrix, B::StiefelLieAlgHorMatrix) = B + A - -Base.:*(α::Real, A::StiefelLieAlgHorMatrix) = A * α - -function Base.zeros(::Type{StiefelLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - StiefelLieAlgHorMatrix( - zeros(SkewSymMatrix{T}, n), - zeros(T, N-n, n), - N, - n - ) -end - -function Base.zeros(::Type{StiefelLieAlgHorMatrix}, N::Integer, n::Integer) - StiefelLieAlgHorMatrix( - zeros(SkewSymMatrix, n), - zeros(N-n, n), - N, - n - ) -end - -function Base.zeros(backend::KernelAbstractions.Backend, ::Type{StiefelLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - StiefelLieAlgHorMatrix( - zeros(backend, SkewSymMatrix{T}, n), - KernelAbstractions.zeros(backend, T, N-n, n), N, n) -end - - -Base.similar(A::StiefelLieAlgHorMatrix, dims::Union{Integer, AbstractUnitRange}...) = zeros(StiefelLieAlgHorMatrix{eltype(A)}, dims...) -Base.similar(A::StiefelLieAlgHorMatrix) = zeros(StiefelLieAlgHorMatrix{eltype(A)}, A.N, A.n) - -function Base.copyto!(A::StiefelLieAlgHorMatrix, B::StiefelLieAlgHorMatrix) - copyto!(A.A, B.A) - copyto!(A.B, B.B) - A -end - -function Base.rand(rng::Random.AbstractRNG, backend::KernelAbstractions.Backend, ::Type{StiefelLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - B = KernelAbstractions.allocate(backend, T, N-n, n) - rand!(rng, B) - StiefelLieAlgHorMatrix(rand(rng, backend, SkewSymMatrix{T}, n), B, N, n) -end - -function Base.rand(backend::KernelAbstractions.Backend, type::Type{StiefelLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - rand(Random.default_rng(), backend, type, N, n) -end - -function Base.rand(rng::Random.AbstractRNG, ::Type{StiefelLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - StiefelLieAlgHorMatrix(rand(rng, SkewSymMatrix{T}, n), rand(rng, T, N-n, n), N, n) -end - -function Base.rand(rng::Random.AbstractRNG, ::Type{StiefelLieAlgHorMatrix}, N::Integer, n::Integer) - StiefelLieAlgHorMatrix(rand(rng, SkewSymMatrix, n), rand(rng, N-n, n), N, n) -end - -function Base.rand(::Type{StiefelLieAlgHorMatrix{T}}, N::Integer, n::Integer) where T - rand(Random.default_rng(), StiefelLieAlgHorMatrix{T}, N, n) -end - -function Base.rand(::Type{StiefelLieAlgHorMatrix}, N::Integer, n::Integer) - rand(Random.default_rng(), StiefelLieAlgHorMatrix, N, n) -end - -function scalar_add(A::StiefelLieAlgHorMatrix, δ::Real) - StiefelLieAlgHorMatrix(scalar_add(A.A, δ), A.B .+ δ, A.N, A.n) -end - -#define these functions more generally! (maybe make a fallback script!!) -function ⊙²(A::StiefelLieAlgHorMatrix) - StiefelLieAlgHorMatrix(⊙²(A.A), A.B.^2, A.N, A.n) -end -function racᵉˡᵉ(A::StiefelLieAlgHorMatrix) - StiefelLieAlgHorMatrix(racᵉˡᵉ(A.A), sqrt.(A.B), A.N, A.n) -end -function /ᵉˡᵉ(A::StiefelLieAlgHorMatrix, B::StiefelLieAlgHorMatrix) - StiefelLieAlgHorMatrix(/ᵉˡᵉ(A.A, B.A), A.B./B.B, A.N, A.n) -end - -function LinearAlgebra.mul!(C::StiefelLieAlgHorMatrix, A::StiefelLieAlgHorMatrix, α::Real) - mul!(C.A, A.A, α) - mul!(C.B, A.B, α) -end -LinearAlgebra.mul!(C::StiefelLieAlgHorMatrix, α::Real, A::StiefelLieAlgHorMatrix) = mul!(C, A, α) -LinearAlgebra.rmul!(C::StiefelLieAlgHorMatrix, α::Real) = mul!(C, C, α) - -@doc raw""" - vec(A::StiefelLieAlgHorMatrix) - -Vectorize `A`. - -# Examples - -```jldoctest -using GeometricMachineLearning - -A = SkewSymMatrix([1, ], 2) -B = [2 3; ] -B̄ = StiefelLieAlgHorMatrix(A, B, 3, 2) -B̄ |> vec - -# output - -vcat(1-element Vector{Int64}, 2-element Vector{Int64}): - 1 - 2 - 3 -``` - -# Implementation - -This is using `Vcat` from the package `LazyArrays`. -""" -function Base.vec(A::StiefelLieAlgHorMatrix) - LazyArrays.Vcat(vec(A.A), vec(A.B)) -end - -function StiefelLieAlgHorMatrix(V::AbstractVector, N::Int, n::Int) - # length of skew-symmetric matrix - skew_sym_size = n*(n-1)÷2 - # size of matrix component - matrix_size = (N-n)*n - @assert length(V) == skew_sym_size + matrix_size - StiefelLieAlgHorMatrix( - SkewSymMatrix(@view(V[1:skew_sym_size]), n), - reshape(@view(V[(skew_sym_size+1):(skew_sym_size+matrix_size)]), (N-n), n), - N, - n - ) -end - -function Base.zero(B::StiefelLieAlgHorMatrix) - StiefelLieAlgHorMatrix( - zero(B.A), - zero(B.B), - B.N, - B.n - ) -end - -function networkbackend(B::StiefelLieAlgHorMatrix) - networkbackend(B.B) -end - -# assign funciton; also implement this for other arrays! -function assign!(B::StiefelLieAlgHorMatrix{T}, C::StiefelLieAlgHorMatrix{T}) where T - assign!(B.A, C.A) - assign!(B.B, C.B) - - nothing -end - -function Base.copy(B::StiefelLieAlgHorMatrix) - StiefelLieAlgHorMatrix( - copy(B.A), - copy(B.B), - B.N, - B.n - ) -end - -# fallback -> put this somewhere else! -function assign!(A::AbstractArray, B::AbstractArray) - A .= B - - nothing -end - -function Base.one(B::StiefelLieAlgHorMatrix{T}) where T - backend = networkbackend(B) - oneB = KernelAbstractions.zeros(backend, T, B.N, B.N) - write_ones! = write_ones_kernel!(backend) - write_ones!(oneB; ndrange = B.N) - - oneB -end - -function _round(B::StiefelLieAlgHorMatrix; kwargs...) - StiefelLieAlgHorMatrix( - _round(B.A; kwargs...), - _round(B.B; kwargs...), - B.N, - B.n - ) -end - -# fills the *storage*; see the comment on `fill!(::SkewSymMatrix, ::Any)` -Base.fill!(A::StiefelLieAlgHorMatrix, val) = (fill!(A.A, val); fill!(A.B, val); A) diff --git a/src/arrays/stiefel_projection.jl b/src/arrays/stiefel_projection.jl deleted file mode 100644 index 5633fd76a..000000000 --- a/src/arrays/stiefel_projection.jl +++ /dev/null @@ -1,84 +0,0 @@ -@doc raw""" - StiefelProjection(backend, T, N, n) - -Make a matrix of the form ``\begin{bmatrix} \mathbb{I} & \mathbb{O} \end{bmatrix}^T`` for a specific backend and data type. - -An array that essentially does `vcat(I(n), zeros(N-n, n))` with GPU support. - -# Extended help - -An instance of `StiefelProjection` should technically also belong to [`StiefelManifold`](@ref). -""" -struct StiefelProjection{T, AT} <: AbstractMatrix{T} - N::Integer - n::Integer - A::AT - function StiefelProjection(backend, T::Type, N::Integer, n::Integer) - A = KernelAbstractions.zeros(backend, T, N, n) - assign_ones_for_stiefel_projection! = assign_ones_for_stiefel_projection_kernel!(backend) - assign_ones_for_stiefel_projection!(A, ndrange=n) - new{T, typeof(A)}(N,n, A) - end -end - -@doc raw""" - StiefelProjection(A::AbstractMatrix) - -Extract necessary information from `A` and build an instance of `StiefelProjection`. - -Necessary information here referes to the backend, the data type and the size of the matrix. -""" -function StiefelProjection(A::AbstractMatrix{T}) where T - StiefelProjection(networkbackend(A), T, size(A)...) -end - -@doc raw""" - StiefelProjection(B::AbstractLieAlgHorMatrix) - -Extract necessary information from `B` and build an instance of `StiefelProjection`. - -Necessary information here referes to the backend, the data type and the size of the matrix. - -The size is queried through `B.N` and `B.n`. - -# Examples - -```jldoctest -using GeometricMachineLearning - -B₁ = rand(StiefelLieAlgHorMatrix, 5, 2) -B₂ = rand(GrassmannLieAlgHorMatrix, 5, 2) -E = [1. 0.; 0. 1.; 0. 0.; 0. 0.; 0. 0.] - -StiefelProjection(B₁) ≈ StiefelProjection(B₂) ≈ E - -# output - -true -``` -""" -function StiefelProjection(B::AbstractLieAlgHorMatrix{T}) where T - StiefelProjection(networkbackend(B), T, B.N, B.n) -end - -@kernel function assign_ones_for_stiefel_projection_kernel!(A::AbstractArray{T}) where T - i = @index(Global) - A[i, i] = one(T) -end - -StiefelProjection(N::Integer, n::Integer, T::Type=Float64) = StiefelProjection(CPU(), T, N, n) - -StiefelProjection(T::Type, N::Integer, n::Integer) = StiefelProjection(N, n, T) - -Base.size(E::StiefelProjection) = (E.N, E.n) -Base.getindex(E::StiefelProjection, i, j) = getindex(E.A, i, j) -Base.:+(E::StiefelProjection, A::AbstractMatrix) = E.A + A -Base.:+(A::AbstractMatrix, E::StiefelProjection) = +(E, A) -Base.vcat(A::AbstractVecOrMat{T}, E::StiefelProjection{T}) where {T<:Number} = vcat(A, E.A) -Base.vcat(E::StiefelProjection{T}, A::AbstractVecOrMat{T}) where {T<:Number} = vcat(E.A, A) -Base.hcat(A::AbstractVecOrMat{T}, E::StiefelProjection{T}) where {T<:Number} = hcat(A, E.A) -Base.hcat(E::StiefelProjection{T}, A::AbstractVecOrMat{T}) where {T<:Number} = hcat(E.A, A) - -function networkbackend(E::StiefelProjection) - networkbackend(E.A) -end \ No newline at end of file diff --git a/src/arrays/symmetric.jl b/src/arrays/symmetric.jl deleted file mode 100644 index 458a99c78..000000000 --- a/src/arrays/symmetric.jl +++ /dev/null @@ -1,310 +0,0 @@ -@doc raw""" - SymmetricMatrix(S::AbstractVector, n::Integer) - -Instantiate a symmetric matrix with information stored in vector `S`. - -A `SymmetricMatrix` ``A`` is a matrix ``A^T = A``. - -Internally the `struct` saves a vector ``S`` of size ``n(n+1)\div2``. The conversion is done the following way: -```math -[A]_{ij} = \begin{cases} S[( (i-1) i ) \div 2 + j] & \text{if $i\geq{}j$}\\ - S[( (j-1) j ) \div 2 + i] & \text{else}. \end{cases} -``` - -So ``S`` stores a string of vectors taken from ``A``: ``S = [\tilde{a}_1, \tilde{a}_2, \ldots, \tilde{a}_n]`` with ``\tilde{a}_i = [[A]_{i1},[A]_{i2},\ldots,[A]_{ii}]``. - -Also see [`SkewSymMatrix`](@ref), [`LowerTriangular`](@ref) and [`UpperTriangular`](@ref). - -# Examples -```jldoctest -using GeometricMachineLearning -S = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] -SymmetricMatrix(S, 4) - -# output - -4×4 SymmetricMatrix{Int64, Vector{Int64}}: - 1 2 4 7 - 2 3 5 8 - 4 5 6 9 - 7 8 9 10 -``` -""" -mutable struct SymmetricMatrix{T, AT <: AbstractVector{T}} <: AbstractMatrix{T} - S::AT - n::Int - - function SymmetricMatrix(S::AbstractVector, n::Integer) - @assert length(S) == n*(n+1)÷2 - new{eltype(S),typeof(S)}(S, n) - end -end - -@doc raw""" - SymmetricMatrix(A::AbstractMatrix) - -Perform a projection and store the matrix in an efficient way (as a vector with ``n(n+1)/2`` entries). - -If the constructor is called with a matrix as input it returns a symmetric matrix via the *projection*: -```math -A \mapsto \frac{1}{2}(A + A^T). -``` - -# Examples -```jldoctest -using GeometricMachineLearning -M = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16] -SymmetricMatrix(M) - -# output - -4×4 SymmetricMatrix{Float64, Vector{Float64}}: - 1.0 3.5 6.0 8.5 - 3.5 6.0 8.5 11.0 - 6.0 8.5 11.0 13.5 - 8.5 11.0 13.5 16.0 -``` - -# Extended help - -Note that the constructor is designed in such a way that it always returns matrices of type `SymmetricMatrix{<:AbstractFloat}` when called with a matrix, even if this matrix is of type `AbstractMatrix{<:Integer}`. - -If the user wishes to allocate a matrix `SymmetricMatrix{<:Integer}` then call - -``julia -SymmetricMatrix(::AbstractVector, n::Integer) -``` - -Note that this is different from [`LowerTriangular`](@ref) and [`UpperTriangular`](@ref) as no porjection takes place there. -""" -function SymmetricMatrix(A::AbstractMatrix{T}) where {T} - S = map_to_S(A) - SymmetricMatrix(S, size(A, 1)) -end - -# I'm not 100% sure this is the best solution (needed for broadcasting operations ...) -function Base.setindex!(A::SymmetricMatrix{T}, val::T, i::Int, j::Int) where T - if i ≥ j - A.S[i * (i-1)÷2 + j] = val - else - A.S[j * (j-1)÷2 + i] = val - end -end - -@kernel function assign_S_val_kernel!(S, A_sym, i) - j = @index(Global) - S[i * (i-1)÷2 + j] = A_sym[i, j] -end - -function map_to_S(A::AbstractMatrix{T}) where {T <: Number} - n = size(A, 1) - @assert size(A, 2) == n - A_sym = T(.5)*(A + A') - backend = networkbackend(A) - S = KernelAbstractions.zeros(backend, T, n*(n+1)÷2) - assign_S_val! = assign_S_val_kernel!(backend) - for i in 1:n - assign_S_val!(S, A_sym, i, ndrange=i) - end - S -end - -function map_to_S(A::AbstractMatrix{T}) where {T <: Integer} - Float = T == Int64 ? Float64 : Float32 - map_to_S(Float.(A)) -end - -function LinearAlgebra.Adjoint(A::SymmetricMatrix) - A -end - -function Base.zero(A::SymmetricMatrix) - SymmetricMatrix(zero(A.S), A.n) -end - -# `similar` has to preserve the type: the optimizer caches allocate their scratch arrays with it and -# then require every one of them to have the same type as the parameter. The generic -# `AbstractArray` fallback returns a dense `Matrix` and makes the cache constructors inapplicable. -Base.similar(A::SymmetricMatrix) = SymmetricMatrix(similar(A.S), A.n) - -# note that this fills the *storage*, i.e. `fill!(A, val)` gives a matrix whose off-diagonal entries -# are `val` and whose diagonal is `val` as well (the diagonal is part of `S` for a symmetric matrix). -Base.fill!(A::SymmetricMatrix, val) = (fill!(A.S, val); A) - -function Base.getindex(A::SymmetricMatrix,i::Int,j::Int) - if i ≥ j - A.S[((i-1)*i)÷2+j] - else - A.S[(j-1)*j÷2+i] - end -end - -Base.parent(A::SymmetricMatrix) = A.S -Base.size(A::SymmetricMatrix) = (A.n,A.n) - -function Base.:+(A::SymmetricMatrix, B::SymmetricMatrix) - @assert A.n == B.n - SymmetricMatrix(A.S + B.S, A.n) -end - -function add!(C::SymmetricMatrix, A::SymmetricMatrix, B::SymmetricMatrix) - @assert A.n == B.n == C.n - add!(C.S, A.S, B.S) -end - -function Base.:-(A::SymmetricMatrix, B::SymmetricMatrix) - @assert A.n == B.n - SymmetricMatrix(A.S - B.S, A.n) -end - - -function Base.:-(A::SymmetricMatrix) - SymmetricMatrix(-A.S, A.n) -end - -function Base.:*(A::SymmetricMatrix, α::Real) - SymmetricMatrix(α*A.S, A.n) -end - -Base.:*(α::Real, A::SymmetricMatrix) = A*α - -function Base.zeros(::Type{SymmetricMatrix{T}}, n::Int) where T - SymmetricMatrix(zeros(T, n*(n+1)÷2), n) -end - -function Base.zeros(::Type{SymmetricMatrix}, n::Int) - SymmetricMatrix(zeros(n*(n+1)÷2), n) -end - -function Base.rand(rng::Random.AbstractRNG, ::Type{SymmetricMatrix{T}}, n::Int) where T - SymmetricMatrix(rand(rng, T, n*(n+1)÷2),n) -end - -function Base.rand(rng::Random.AbstractRNG, ::Type{SymmetricMatrix}, n::Int) - SymmetricMatrix(rand(rng, n*(n+1)÷2), n) -end - -#TODO: make defaults when no rng is specified!!! (prbabaly rng ← Random.default_rng()) -function Base.rand(type::Type{SymmetricMatrix{T}}, n::Integer) where T - rand(Random.default_rng(), type, n) -end - -function Base.rand(type::Type{SymmetricMatrix}, n::Integer) - rand(Random.default_rng(), type, n) -end - -#these are Adam operations: -function scalar_add(A::SymmetricMatrix, δ::Real) - SymmetricMatrix(A.S .+ δ, A.n) -end - -#element-wise squares and square root (for Adam) -function ⊙²(A::SymmetricMatrix) - SymmetricMatrix(A.S.^2, A.n) -end -function racᵉˡᵉ(A::SymmetricMatrix) - SymmetricMatrix(sqrt.(A.S), A.n) -end -function /ᵉˡᵉ(A::SymmetricMatrix, B::SymmetricMatrix) - @assert A.n == B.n - SymmetricMatrix(A.S ./ B.S, A.n) -end - -function LinearAlgebra.mul!(C::SymmetricMatrix, A::SymmetricMatrix, α::Real) - mul!(C.S, A.S, α) -end -LinearAlgebra.mul!(C::SymmetricMatrix, α::Real, A::SymmetricMatrix) = mul!(C, A, α) -LinearAlgebra.rmul!(C::SymmetricMatrix, α::Real) = mul!(C, C, α) - -@kernel function symmetric_mat_mul_kernel!(C::AbstractMatrix{T}, S::AbstractVector{T}, B::AbstractMatrix{T}, n) where T - i, j = @index(Global, NTuple) - - tmp_sum = zero(T) - for k = 1:i - tmp_sum += S[((i-1)*i)÷2+k] * B[k, j] - end - for k = (i+1):n - tmp_sum += S[((k-1)*k)÷2+i] * B[k, j] - end - C[i, j] = tmp_sum -end - -function LinearAlgebra.mul!(C::AbstractMatrix, A::SymmetricMatrix, B::AbstractMatrix) - @assert A.n == size(B, 1) - @assert size(B, 2) == size(C, 2) - @assert A.n == size(C, 1) - backend = networkbackend(A.S) - symmetric_mat_mul! = symmetric_mat_mul_kernel!(backend) - symmetric_mat_mul!(C, A.S, B, A.n, ndrange=size(C)) -end - -@kernel function symmetric_vector_mul_kernel!(c::AbstractVector{T}, S::AbstractVector{T}, b::AbstractVector{T}, n) where T - i = @index(Global) - - tmp_sum = zero(T) - for k = 1:i - tmp_sum += S[((i-1)*i)÷2+k] * b[k] - end - for k = (i+1):n - tmp_sum += S[((k-1)*k)÷2+i] * b[k] - end - c[i] = tmp_sum -end - -function LinearAlgebra.mul!(c::AbstractVector, A::SymmetricMatrix, b::AbstractVector) - @assert A.n == length(c) == length(b) - backend = networkbackend(A.S) - symmetric_vector_mul! = symmetric_vector_mul_kernel!(backend) - symmetric_vector_mul!(c, A.S, b, A.n, ndrange=size(c)) -end - -function Base.:*(A::SymmetricMatrix{T}, B::AbstractMatrix{T}) where T - backend = networkbackend(A.S) - C = KernelAbstractions.allocate(backend, T, A.n, size(B, 2)) - LinearAlgebra.mul!(C, A, B) - C -end - -Base.:*(B::AbstractMatrix{T}, A::SymmetricMatrix{T}) where T = (A * B')' - -function Base.:*(A::SymmetricMatrix{T}, B::SymmetricMatrix{T}) where T - A * (B * one(B)) -end - -function Base.:*(A::SymmetricMatrix{T}, b::AbstractVector{T}) where T - backend = networkbackend(A.S) - c = KernelAbstractions.allocate(backend, T, A.n) - LinearAlgebra.mul!(c, A, b) - c -end - -function Base.one(A::SymmetricMatrix{T}) where T - backend = networkbackend(A.S) - unit_matrix = KernelAbstractions.zeros(backend, T, A.n, A.n) - write_ones! = write_ones_kernel!(backend) - write_ones!(unit_matrix, ndrange=A.n) - unit_matrix -end - -function assign!(B::SymmetricMatrix{T}, C::SymmetricMatrix{T}) where T - B.S .= C.S - - nothing -end - -function Base.copy(A::SymmetricMatrix) - SymmetricMatrix(copy(A.S), A.n) -end - -Base.vec(A::SymmetricMatrix) = A.S - -function Base.copyto!(A::SymmetricMatrix{T}, B::SymmetricMatrix{T}) where T - A.S .= B.S - - nothing -end - -# define routines for generalizing ChainRulesCore to SymmetricMatrix -ChainRulesCore.ProjectTo(A::SymmetricMatrix) = ProjectTo{SymmetricMatrix}(; symmetric=ProjectTo(A.S)) -(project::ProjectTo{SymmetricMatrix})(dA::AbstractMatrix) = SymmetricMatrix(project.symmetric(map_to_S(dA)), size(dA, 2)) -(project::ProjectTo{SymmetricMatrix})(dA::SymmetricMatrix) = SymmetricMatrix(project.symmetric(dA.S), dA.n) \ No newline at end of file diff --git a/src/arrays/triangular.jl b/src/arrays/triangular.jl deleted file mode 100644 index ab07a55d7..000000000 --- a/src/arrays/triangular.jl +++ /dev/null @@ -1,164 +0,0 @@ -@doc raw""" - AbstractTriangular - -See [`UpperTriangular`](@ref) and [`LowerTriangular`](@ref). -""" -abstract type AbstractTriangular{T} <: AbstractMatrix{T} end - -Base.parent(A::AbstractTriangular) = A.S -Base.size(A::AbstractTriangular) = (A.n, A.n) - -function Base.:+(A::AT, B::AT) where AT <: AbstractTriangular - @assert A.n == B.n - AT(A.S + B.S, A.n) -end - -function add!(C::AT, A::AT, B::AT) where AT <: AbstractTriangular - @assert A.n == B.n == C.n - add!(C.S, A.S, B.S) -end - -function Base.:-(A::AT, B::AT) where AT <: AbstractTriangular - @assert A.n == B.n - AT(A.S - B.S, A.n) -end - -function Base.:-(A::AT) where AT <: AbstractTriangular - AT(-A.S, A.n) -end - -function Base.:*(A::AT, α::Real) where AT <: AbstractTriangular - AT(α * A.S, A.n) -end - -Base.:*(α::Real, A::AT) where AT <: AbstractTriangular = A * α - -function Base.zeros(backend::KernelAbstractions.Backend, ::Type{AT}, n::Int) where {T, AT <: AbstractTriangular{T}} - # nameof converts AT to :UpperTriangular or ::LowerTriangular - eval(nameof(AT))(KernelAbstractions.zeros(backend, T, n*(n-1)÷2), n) -end - -function Base.zeros(::Type{AT}, n::Int) where {T, AT <: AbstractTriangular{T}} - zeros(CPU(), AT, n) -end - -function Base.rand(rng::AbstractRNG, backend::KernelAbstractions.Backend, ::Type{AT}, n::Integer) where {T, AT <: AbstractTriangular{T}} - S = KernelAbstractions.allocate(backend, T, n*(n-1)÷2) - Random.rand!(rng, S) - eval(nameof(AT))(S, n) -end - -function Base.rand(rng::Random.AbstractRNG, type::Type{AT}, n::Int) where {T, AT <: AbstractTriangular{T}} - rand(rng, CPU(), type, n) -end - -function Base.rand(type::Type{AT}, n::Integer) where {T, AT <: AbstractTriangular{T}} - rand(Random.default_rng(), type, n) -end - -function Base.rand(::Type{AT}, n::Integer) where {AT <: AbstractTriangular} - rand(AT{Float64}, n) -end - -function Base.rand(backend::KernelAbstractions.Backend, type::Type{AT}, n::Integer) where {T, AT <: AbstractTriangular{T}} - rand(Random.default_rng(), backend, type, n) -end - -# these are Adam operations: -function scalar_add(A::AT, δ::Real) where {T, AT <: AbstractTriangular{T}} - AT(A.S .+ δ, A.n) -end - -#element-wise squares and square root (for Adam) -function ⊙²(A::AT) where AT <: AbstractTriangular - AT(A.S.^2, A.n) -end -function racᵉˡᵉ(A::AT) where AT <: AbstractTriangular - AT(sqrt.(A.S), A.n) -end -function /ᵉˡᵉ(A::AT, B::AT) where AT <: AbstractTriangular - @assert A.n == B.n - AT(A.S ./ B.S, A.n) -end - -function LinearAlgebra.mul!(C::AT, A::AT, α::Real) where AT <: AbstractTriangular - mul!(C.S, A.S, α) -end -LinearAlgebra.mul!(C::AT, α::Real, A::AT) where AT <: AbstractTriangular = mul!(C, A, α) -LinearAlgebra.rmul!(C::AT, α::Real) where AT <: AbstractTriangular = mul!(C, C, α) - -function Base.one(A::AbstractTriangular{T}) where T - backend = networkbackend(A.S) - unit_matrix = KernelAbstractions.zeros(backend, T, A.n, A.n) - write_ones! = write_ones_kernel!(backend) - write_ones!(unit_matrix, ndrange=A.n) - unit_matrix -end - -# the first matrix is multiplied onto A2 in order for it to not be SkewSymMatrix! -function Base.:*(A1::AbstractTriangular{T}, A2::AbstractTriangular{T}) where T - A1 * (A2 * one(A2)) -end - -@doc raw""" - vec(A::AbstractTriangular) - -Return the associated vector to ``A``. - -# Examples - -```jldoctest -using GeometricMachineLearning - -M = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16] -LowerTriangular(M) |> vec - -# output - -6-element Vector{Int64}: - 5 - 9 - 10 - 13 - 14 - 15 -``` -""" -function Base.vec(A::AbstractTriangular) - A.S -end - -function Base.zero(A::AT) where AT <: AbstractTriangular - AT(zero(A.S), A.n) -end - -# see the comment on `similar(::SymmetricMatrix)` -Base.similar(A::AT) where {AT <: AbstractTriangular} = AT(similar(A.S), A.n) - -# this fills the *storage*, so the entries outside the stored triangle stay zero -Base.fill!(A::AbstractTriangular, val) = (fill!(A.S, val); A) - -function networkbackend(A::AbstractTriangular) - networkbackend(A.S) -end - -function assign!(B::AT, C::AT) where AT <: AbstractTriangular - B.S .= C.S -end - -function Base.copy(A::AT) where AT <: AbstractTriangular - AT(copy(A.S), A.n) -end - -function Base.copyto!(A::AbstractTriangular, B::AbstractTriangular) - A.S .= B.S - nothing -end - -function Base.:*(A::AbstractTriangular, b::AbstractVector{T}) where T - A * reshape(b, length(b), 1) -end - -function Base.:*(B::AbstractMatrix{T}, A::AbstractTriangular{T}) where T - (A' * B')' -end \ No newline at end of file diff --git a/src/arrays/upper_triangular.jl b/src/arrays/upper_triangular.jl deleted file mode 100644 index 07c2482d2..000000000 --- a/src/arrays/upper_triangular.jl +++ /dev/null @@ -1,105 +0,0 @@ -@doc raw""" - UpperTriangular(S::AbstractVector, n::Int) - -Build an upper-triangular matrix from a vector. - -An upper-triangular matrix is an ``n\times{}n`` matrix that has zeros on the diagonal and on the lower triangular. - -The data are stored in a vector ``S`` similarly to other matrices. See [`LowerTriangular`](@ref), [`SkewSymMatrix`](@ref) and [`SymmetricMatrix`](@ref). - -The struct two fields: `S` and `n`. The first stores all the entries of the matrix in a sparse fashion (in a vector) and the second is the dimension ``n`` for ``A\in\mathbb{R}^{n\times{}n}``. - -# Examples -```jldoctest -using GeometricMachineLearning -S = [1, 2, 3, 4, 5, 6] -UpperTriangular(S, 4) - -# output - -4×4 UpperTriangular{Int64, Vector{Int64}}: - 0 1 2 4 - 0 0 3 5 - 0 0 0 6 - 0 0 0 0 -``` -""" -mutable struct UpperTriangular{T, AT <: AbstractVector{T}} <: AbstractTriangular{T} - S::AT - n::Int -end - -@doc raw""" - UpperTriangular(A::AbstractMatrix) - -Build an upper-triangular matrix from a matrix. - -This is done by taking the upper right of that matrix. - -# Examples -```jldoctest -using GeometricMachineLearning -M = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16] -UpperTriangular(M) - -# output - -4×4 UpperTriangular{Int64, Vector{Int64}}: - 0 2 3 4 - 0 0 7 8 - 0 0 0 12 - 0 0 0 0 -``` -""" -function UpperTriangular(S::AbstractMatrix{T}) where {T} - n = size(S, 1) - @assert size(S, 2) == n - S_vec = map_to_up(S) - UpperTriangular(S_vec, n) -end - -function Base.getindex(A::UpperTriangular{T}, i::Int, j::Int) where T - if j == i - return zero(T) - end - if j > i - return A.S[(j-2) * (j-1) ÷ 2 + i] - end - return zero(T) -end - -@kernel function up_mat_mul_kernel!(C::AbstractMatrix{T}, S::AbstractVector{T}, B::AbstractMatrix{T}, n) where T - i, j = @index(Global, NTuple) - - tmp_sum = zero(T) - for k = (i + 1):n - tmp_sum += S[(k - 2) * (k - 1) ÷ 2 + i] * B[k, j] - end - C[i, j] = tmp_sum -end - -function map_to_up(A::AbstractMatrix{T}) where T - n = size(A, 1) - @assert size(A, 2) == n - backend = networkbackend(A) - S = KernelAbstractions.zeros(backend, T, n * (n - 1) ÷ 2) - assign_Skew_val! = assign_Skew_val_kernel!(backend) - for i in 2:n - assign_Skew_val!(S, A', i, ndrange = (i - 1)) - end - S -end - - -# define routines for generalizing ChainRulesCore to UpperTriangular -ChainRulesCore.ProjectTo(A::AT) where AT <: UpperTriangular = ProjectTo{AT}(; triang = ProjectTo(A.S)) -(project::ProjectTo{<:UpperTriangular})(dA::AbstractMatrix) = UpperTriangular(project.triang(map_to_up(dA)), size(dA, 2)) -(project::ProjectTo{<:UpperTriangular})(dA::UpperTriangular) = UpperTriangular(project.triang(dA.S), dA.n) - -function Base.adjoint(A::LowerTriangular) - UpperTriangular(A.S, A.n) -end - -function Base.adjoint(A::UpperTriangular) - LowerTriangular(A.S, A.n) -end \ No newline at end of file diff --git a/src/layers/linear_symplectic_attention.jl b/src/layers/linear_symplectic_attention.jl index 0bad20400..91846e573 100644 --- a/src/layers/linear_symplectic_attention.jl +++ b/src/layers/linear_symplectic_attention.jl @@ -9,7 +9,7 @@ For more information see [`LinearSymplecticAttentionQ`](@ref) and [`LinearSymple # Implementation -The coefficients of a [`LinearSymplecticAttention`](@ref) layer is a [`SymmetricMatrix`](@ref): +The coefficients of a [`LinearSymplecticAttention`](@ref) layer is a [`SymmetricMatrix`](@extref GeometricOptimizers GeometricOptimizers.SymmetricMatrix): ```jldoctest using GeometricMachineLearning diff --git a/src/layers/multi_head_attention.jl b/src/layers/multi_head_attention.jl index 61190ccfc..966e04341 100644 --- a/src/layers/multi_head_attention.jl +++ b/src/layers/multi_head_attention.jl @@ -18,7 +18,7 @@ The optional keyword arguments to `MultiHeadAttention` are: - `add_connection::Bool=true` - `activation::AbstractSoftmax=`[`VectorSoftmax`](@ref). -`Stiefel` indicates whether weights are put on the [`StiefelManifold`](@ref) ``St(\mathrm{dim}, \mathrm{dim}\div\mathrm{n\_heads})``. +`Stiefel` indicates whether weights are put on the [`StiefelManifold`](@extref GeometricOptimizers GeometricOptimizers.StiefelManifold) ``St(\mathrm{dim}, \mathrm{dim}\div\mathrm{n\_heads})``. `add_connection` indicates whether the input is again added to the output. """ diff --git a/src/layers/sympnets.jl b/src/layers/sympnets.jl index ba6e2c151..5122467a3 100644 --- a/src/layers/sympnets.jl +++ b/src/layers/sympnets.jl @@ -64,7 +64,7 @@ See the constructors [`LinearLayerQ`](@ref) and [`LinearLayerP`](@ref). # Implementation -`LinearLayer` uses the custom matrix [`SymmetricMatrix`](@ref) for its weight. +`LinearLayer` uses the custom matrix [`SymmetricMatrix`](@extref GeometricOptimizers GeometricOptimizers.SymmetricMatrix) for its weight. """ struct LinearLayer{M, N, C} <: SympNetLayer{M, N} end @@ -81,7 +81,7 @@ This is equivalent to a left multiplication by the matrix: \mathbb{O} & \mathbb{I} \end{pmatrix}, ``` -where ``A`` is a [`SymmetricMatrix`](@ref). +where ``A`` is a [`SymmetricMatrix`](@extref GeometricOptimizers GeometricOptimizers.SymmetricMatrix). """ const LinearLayerQ{M, N, TA} = LinearLayer{M, N, :Q} @@ -97,7 +97,7 @@ This is equivalent to a left multiplication by the matrix: A & \mathbb{I} \end{pmatrix}, ``` -where ``A`` is a [`SymmetricMatrix`](@ref). +where ``A`` is a [`SymmetricMatrix`](@extref GeometricOptimizers GeometricOptimizers.SymmetricMatrix). """ const LinearLayerP{M, N, TA} = LinearLayer{M, N, :P} diff --git a/src/layers/volume_preserving_feedforward.jl b/src/layers/volume_preserving_feedforward.jl index 561836387..9615810ca 100644 --- a/src/layers/volume_preserving_feedforward.jl +++ b/src/layers/volume_preserving_feedforward.jl @@ -7,7 +7,7 @@ Super-type of [`VolumePreservingLowerLayer`](@ref) and [`VolumePreservingUpperLa x \mapsto \begin{cases} \sigma(Lx + b) & \text{where $L$ is }\mathtt{LowerTriangular}, \\ \sigma(Ux + b) & \text{where $U$ is }\mathtt{UpperTriangular}. \end{cases} ``` -The functor can be applied to a vector, a matrix or a tensor. The special matrices are implemented as [`LowerTriangular`](@ref) and [`UpperTriangular`](@ref). +The functor can be applied to a vector, a matrix or a tensor. The special matrices are implemented as [`LowerTriangular`](@extref GeometricOptimizers GeometricOptimizers.LowerTriangular) and [`UpperTriangular`](@extref GeometricOptimizers GeometricOptimizers.UpperTriangular). """ abstract type VolumePreservingFeedForwardLayer{M, N, bias} <: AbstractExplicitLayer{M, N} end diff --git a/src/manifolds/abstract_manifold.jl b/src/manifolds/abstract_manifold.jl deleted file mode 100644 index f58ee6603..000000000 --- a/src/manifolds/abstract_manifold.jl +++ /dev/null @@ -1,116 +0,0 @@ -@doc raw""" - Manifold <: AbstractMatrix - -A manifold in `GeometricMachineLearning` is a sutype of `AbstractMatrix`. All manifolds are matrix manifolds and therefore stored as matrices. More details can be found in the docstrings for the [`StiefelManifold`](@ref) and the [`GrassmannManifold`](@ref). -""" -abstract type Manifold{T} <: AbstractMatrix{T} end - -@kernel function assign_columns_kernel!(Y::AbstractMatrix{T}, A::AbstractMatrix{T}) where T - i,j = @index(Global, NTuple) - Y[i,j] = A[i,j] -end - -function assign_columns(Q::AbstractMatrix{T}, N::Integer, n::Integer) where T - backend = networkbackend(Q) - Y = KernelAbstractions.allocate(backend, T, N, n) - assign_columns! = assign_columns_kernel!(backend) - assign_columns!(Y, Q, ndrange=size(Y)) - Y -end - -# TODO: check the distribution this is coming from - related to the Haar measure ??? -function Base.rand(::CPU, rng::Random.AbstractRNG, ::Type{MT}, N::Integer, n::Integer) where {T, MT<:Manifold{T}} - @assert N ≥ n - A = randn(rng, T, N, n) - MT{typeof(A)}(assign_columns(typeof(A)(qr!(A).Q), N, n)) -end - -function Base.rand(backend::GPU, rng::Random.AbstractRNG, ::Type{MT}, N::Integer, n::Integer) where {T, MT<:Manifold{T}} - @assert N ≥ n - A = KernelAbstractions.allocate(backend, T, N, n) - Random.randn!(rng, A) - MT{typeof(A)}(assign_columns(typeof(A)(qr!(A).Q), N, n)) -end - -function Base.rand(backend::CPU, rng::Random.AbstractRNG, ::Type{MT}, N::Integer, n::Integer) where MT <: Manifold - rand(backend, rng, MT{Float64}, N, n) -end - -function Base.rand(backend::GPU, rng::Random.AbstractRNG, ::Type{MT}, N::Integer, n::Integer) where MT <: Manifold - rand(backend, rng, MT{Float32}, N, n) -end - -function Base.rand(rng::Random.AbstractRNG, manifold_type::Type{MT}, N::Integer, n::Integer) where MT <: Manifold - rand(CPU(), rng, manifold_type, N, n) -end - -function _round(Y::Manifold; kwargs...) - typeof(Y)(round.(Y.A; kwargs...)) -end - -function Base.broadcast(operation, Y::Manifold) - typeof(Y)(broadcast(operation, Y.A)) -end - -@doc raw""" - rand(backend, manifold_type, N, n) - -Draw random elements for a specific device. - -# Examples - -Random elements of the manifold can be allocated on GPU. Call ... - -```julia -rand(CUDABackend(), StiefelManifold{Float32}, N, n) -``` - -... for drawing elements on a `CUDA` device. -""" -function Base.rand(backend::KernelAbstractions.Backend, manifold_type::Type{MT}, N::Integer, n::Integer) where MT <: Manifold - rand(backend, Random.default_rng(), manifold_type, N, n) -end - -@doc raw""" - rand(manifold_type, N, n) - -Draw random elements from the Stiefel and the Grassmann manifold. - -Because both of these manifolds are compact spaces we can sample them uniformly [mezzadri2006generate](@cite). - -# Examples -When we call ... - -```jldoctest -using GeometricMachineLearning -using GeometricMachineLearning: _round # hide -import Random -Random.seed!(123) - -N, n = 5, 3 -Y = rand(StiefelManifold{Float32}, N, n) -_round(Y; digits = 5) # hide - -# output - -5×3 StiefelManifold{Float32, Matrix{Float32}}: - -0.27575 0.32991 0.77275 - -0.62485 -0.33224 -0.0686 - -0.69333 0.36724 -0.18988 - -0.09295 -0.73145 0.46064 - 0.2102 0.33301 0.38717 -``` - -... the sampling is done by first allocating a random matrix of size ``N\times{}n`` via `Y = randn(Float32, N, n)`. - -We then perform a QR decomposition `Q, R = qr(Y)` with the `qr` function from the `LinearAlgebra` package (this is using Householder reflections internally). - -The final output are then the first `n` columns of the `Q` matrix. -""" -function Base.rand(manifold_type::Type{MT}, N::Integer, n::Integer) where MT <: Manifold - rand(Random.default_rng(), manifold_type, N, n) -end - -Base.size(A::Manifold) = size(A.A) -Base.parent(A::Manifold) = A.A -Base.getindex(A::Manifold, i::Int, j::Int) = A.A[i,j] \ No newline at end of file diff --git a/src/manifolds/grassmann_manifold.jl b/src/manifolds/grassmann_manifold.jl deleted file mode 100644 index fc77577ae..000000000 --- a/src/manifolds/grassmann_manifold.jl +++ /dev/null @@ -1,236 +0,0 @@ -""" - GrassmannManifold <: Manifold - -The `GrassmannManifold` is based on the [`StiefelManifold`](@ref). -""" -mutable struct GrassmannManifold{T, AT <: AbstractMatrix{T}} <: Manifold{T} - A::AT -end - -@doc raw""" - rgrad(Y::GrassmannManifold, ∇L::AbstractMatrix) - -Compute the Riemannian gradient for the Grassmann manifold at `Y` based on `∇L`. - -Here ``Y`` is a representation of ``\mathrm{span}(Y)\in{}Gr(n, N)`` and ``\nabla{}L\in\mathbb{R}^{N\times{}n}`` is the Euclidean gradient. - -This gradient has the property that it is orthogonal to the space spanned by ``Y``. - -The precise form of the mapping is: -```math -\mathtt{rgrad}(Y, \nabla{}L) \mapsto \nabla{}L - YY^T\nabla{}L. -``` - -Note the property ``Y^T\mathrm{rgrad}(Y, \nabla{}L) = \mathbb{O}.`` - -Also see [`rgrad(::StiefelManifold, ::AbstractMatrix)`](@ref). - -# Examples - -```jldoctest -using GeometricMachineLearning - -Y = GrassmannManifold([1 0 ; 0 1 ; 0 0; 0 0]) -Δ = [1 2; 3 4; 5 6; 7 8] -rgrad(Y, Δ) - -# output - -4×2 Matrix{Int64}: - 0 0 - 0 0 - 5 6 - 7 8 -``` -""" -function rgrad(Y::GrassmannManifold, ∇L::AbstractMatrix) - ∇L - Y * (Y' * ∇L) -end - -@doc raw""" - metric(Y::GrassmannManifold, Δ₁::AbstractMatrix, Δ₂::AbstractMatrix) - -Compute the metric for vectors `Δ₁` and `Δ₂` at `Y`. - -The representation of the Grassmann manifold is realized as a *quotient space of the Stiefel manifold*. - -The metric for the Grassmann manifold is: - -```math -g^{Gr}_Y(\Delta_1, \Delta_2) = g^{St}_Y(\Delta_1, \Delta_2) = \mathrm{Tr}(\Delta_1^T (\mathbb{I} - Y Y^T) \Delta_2) = \mathrm{Tr}(\Delta_1^T \Delta_2), -``` -where we used that ``Y^T\Delta_i`` for ``i = 1, 2.`` -""" -function metric(::GrassmannManifold, Δ₁::AbstractMatrix, Δ₂::AbstractMatrix) - LinearAlgebra.tr(Δ₁' * Δ₂) -end - -@doc raw""" - global_section(Y::GrassmannManifold) - -Compute a matrix of size ``N\times(N-n)`` whose columns are orthogonal to the columns in `Y`. - -The method `global_section` for the Grassmann manifold is equivalent to that for the [`StiefelManifold`](@ref) (we represent the Grassmann manifold as an embedding in the Stiefel manifold). - -See the documentation for [`global_section(Y::StiefelManifold{T}) where T`](@ref). -""" -function global_section(Y::GrassmannManifold{T}) where T - N, n = size(Y) - backend = networkbackend(Y) - A = KernelAbstractions.allocate(backend, T, N, N-n) - randn!(A) - A = A - Y.A * (Y.A' * A) - typeof(Y.A)(qr!(A).Q) -end - -GeometricOptimizers.global_section(Y::GrassmannManifold) = global_section(Y) - -function GeometricOptimizers.apply_section!( - Y::GrassmannManifold{T}, - λY::GeometricOptimizers.GlobalSection{T, <:GrassmannManifold{T}}, - Y₂::GrassmannManifold{T} -) where T - N, n = size(λY.Y) - @views Y.A .= λY.Y.A * Y₂.A[1:n, :] .+ λY.λ * Y₂.A[(n + 1):N, :] - Y -end - -@doc raw""" - Ω(Y::GrassmannManifold{T}, Δ::AbstractMatrix{T}) where T - -Perform the *canonical horizontal lift* for the Grassmann manifold: - -```math - \Delta \mapsto \Omega^{St}(\Delta), -``` - -where ``\Omega^{St}`` is the canonical horizontal lift for the Stiefel manifold. - -```jldoctest -using GeometricMachineLearning -E = GrassmannManifold(StiefelProjection(5, 2)) -Δ = [0. 0.; 0. 0.; 2. 3.; 4. 5.; 6. 7.] -GeometricMachineLearning.Ω(E, Δ) - -# output - -5×5 SkewSymMatrix{Float64, Vector{Float64}}: - 0.0 -0.0 -2.0 -4.0 -6.0 - 0.0 0.0 -3.0 -5.0 -7.0 - 2.0 3.0 0.0 -0.0 -0.0 - 4.0 5.0 0.0 0.0 -0.0 - 6.0 7.0 0.0 0.0 0.0 -``` -""" -function Ω(Y::GrassmannManifold{T}, Δ::AbstractMatrix{T}) where T - YY = Y * Y' - - ΩSt = 2 * (one(YY) - T(.5) * Y * Y') * Δ * Y' - # E = StiefelProjection(Y) - # SkewSymMatrix(ΩSt - E * E' * ΩSt * E * E') - SkewSymMatrix(ΩSt) -end - -function Base.copyto!(A::GrassmannManifold, B::GrassmannManifold) - A.A .= B.A - nothing -end - -Base.copy(A::GrassmannManifold) = GrassmannManifold(copy(A.A)) -Base.similar(A::GrassmannManifold) = GrassmannManifold(similar(A.A)) - -function Base.zero(Y::GrassmannManifold{T}) where T - N, n = size(Y) - backend = networkbackend(Y.A) - zeros(backend, GrassmannLieAlgHorMatrix{T}, N, n) -end - -function GeometricOptimizers.global_rep( - λY::GeometricOptimizers.GlobalSection{T, <:GrassmannManifold{T}}, - Δ::AbstractMatrix{T} -) where T - N, n = size(λY.Y) - GrassmannLieAlgHorMatrix( - λY.λ' * Δ, - N, n - ) -end - -function GeometricOptimizers.geodesic(Y::GrassmannManifold{T}, Δ::AbstractMatrix{T}) where T - λY = GeometricOptimizers.GlobalSection(Y) - B = GeometricOptimizers.global_rep(λY, Δ) - E = StiefelProjection(B) - expB = GeometricOptimizers.geodesic(B) - GeometricOptimizers.apply_section(λY, GrassmannManifold(expB * E)) -end - -function GeometricOptimizers.cayley(Y::GrassmannManifold{T}, Δ::AbstractMatrix{T}) where T - λY = GeometricOptimizers.GlobalSection(Y) - B = GeometricOptimizers.global_rep(λY, Δ) - E = StiefelProjection(B) - cayleyB = GeometricOptimizers.cayley(B) - GeometricOptimizers.apply_section(λY, GrassmannManifold(cayleyB * E)) -end - -function GeometricOptimizers.update_section!( - Λᵗ::GeometricOptimizers.GlobalSection{T, <:GrassmannManifold{T}}, - Λ⁽ᵗ⁻¹⁾::GeometricOptimizers.GlobalSection{T, <:GrassmannManifold{T}}, - B⁽ᵗ⁻¹⁾::AbstractMatrix{T}, - retraction -) where T - N, n = B⁽ᵗ⁻¹⁾.N, B⁽ᵗ⁻¹⁾.n - expB = retraction(B⁽ᵗ⁻¹⁾) - GeometricOptimizers.apply_section!(expB, Λ⁽ᵗ⁻¹⁾, expB) - Λᵗ.Y.A .= @view expB.A[:, 1:n] - Λᵗ.λ .= @view expB.A[:, (n+1):N] - nothing -end - -@doc raw""" - cayley(B̄::GrassmannLieAlgHorMatrix) - -Compute the Cayley retraction of an element of [`GrassmannLieAlgHorMatrix`](@ref). - -This is equivalent to [`cayley(::StiefelLieAlgHorMatrix)`](@ref) with ``A = \mathbb{O}``. -""" -function GeometricOptimizers.cayley(B::GrassmannLieAlgHorMatrix) - T = eltype(B) - backend = networkbackend(B) - E = StiefelProjection(B) - 𝕆 = KernelAbstractions.zeros(backend, T, B.n, B.n) - 𝕀_small = one(𝕆) - 𝕀_small2 = hcat(vcat(𝕀_small, 𝕆), vcat(𝕆, 𝕀_small)) - 𝕀_big = one(B) - B̂ = hcat(vcat(𝕆, B.B), E) - B̄ = hcat(vcat(𝕀_small, 𝕆), vcat(zero(B.B'), -B.B'))' - GrassmannManifold((𝕀_big + T(0.5) * B̂ * inv(𝕀_small2 - T(0.5) * B̄' * B̂) * B̄') * (𝕀_big + T(0.5) * B)) -end - -@doc raw""" - geodesic(B̄::GrassmannLieAlgHorMatrix) - -Compute the geodesic of an element of [`GrassmannLieAlgHorMatrix`](@ref). - -This is equivalent to [`geodesic(::StiefelLieAlgHorMatrix)`](@ref) with ``A = \mathbb{O}``. -""" -function GeometricOptimizers.geodesic(B::GrassmannLieAlgHorMatrix) - T = eltype(B) - E = StiefelProjection(B) - backend = networkbackend(B) - zero_mat = KernelAbstractions.zeros(backend, T, B.n, B.n) - B̂ = hcat(vcat(zero_mat, B.B), E) - B̄ = hcat(vcat(one(zero_mat), zero_mat), vcat(zero(B.B'), -B.B'))' - GrassmannManifold(one(B) + B̂ * GeometricOptimizers.𝔄(B̂, B̄) * B̄') -end - -function GeometricOptimizers._copyto!( - Λ₁::GeometricOptimizers.GlobalSection{T, <:GrassmannManifold{T}}, - Λ₂::GeometricOptimizers.GlobalSection{T, <:GrassmannManifold{T}} -) where T - copyto!(Λ₁.Y, Λ₂.Y) - copyto!(Λ₁.λ, Λ₂.λ) - Λ₁ -end - -# The elementwise arithmetic GO needs on `GrassmannLieAlgHorMatrix` lives in -# `src/optimizers/go_bridges.jl`, together with the same bridges for the other GML array types. diff --git a/src/manifolds/stiefel_manifold.jl b/src/manifolds/stiefel_manifold.jl deleted file mode 100644 index a16260ff4..000000000 --- a/src/manifolds/stiefel_manifold.jl +++ /dev/null @@ -1,380 +0,0 @@ -@doc raw""" - StiefelManifold <: Manifold - -An implementation of the Stiefel manifold [hairer2006geometric](@cite). The Stiefel manifold is the collection of all matrices ``Y\in\mathbb{R}^{N\times{}n}`` whose columns are orthonormal, i.e. - -```math - St(n, N) = \{Y: Y^TY = \mathbb{I}_n \}. -``` - -The Stiefel manifold can be shown to have manifold structure (as the name suggests) and this is heavily used in `GeometricMachineLearning`. It is further a compact space. -More information can be found in the docstrings for [`rgrad(::StiefelManifold, ::AbstractMatrix)`](@ref) and [`metric(::StiefelManifold, ::AbstractMatrix, ::AbstractMatrix)`](@ref). -""" -mutable struct StiefelManifold{T, AT <: AbstractMatrix{T}} <: Manifold{T} - A::AT -end - -Base.:*(Y::StiefelManifold, B::AbstractMatrix) = Y.A*B -Base.:*(B::AbstractMatrix, Y::StiefelManifold) = B*Y.A - -function Base.:*(Y::Adjoint{T, StiefelManifold{T, AT}}, B::AbstractMatrix) where {T, AT<:AbstractMatrix{T}} - Y.parent.A'*B -end - -function Base.:*(Y::Adjoint{T, StiefelManifold{T, AT}}, B::StiefelManifold{T, AT}) where {T, AT<:AbstractMatrix{T}} - Y.parent.A' * B.A -end - -function Base.:*(Y::Adjoint{T, ST}, B::ST) where {T, AT<:AbstractMatrix{T}, ST<:StiefelManifold{T, AT}} - Y.parent.A' * B.A -end - -@doc raw""" - rgrad(Y::StiefelManifold, ∇L::AbstractMatrix) - -Compute the Riemannian gradient for the Stiefel manifold at `Y` based on `∇L`. - -Here ``Y\in{}St(N,n)`` and ``\nabla{}L\in\mathbb{R}^{N\times{}n}`` is the Euclidean gradient. - -The function computes the Riemannian gradient with respect to the canonical metric: -[`metric(::StiefelManifold, ::AbstractMatrix, ::AbstractMatrix)`](@ref). - -The precise form of the mapping is: -```math -\mathtt{rgrad}(Y, \nabla{}L) \mapsto \nabla{}L - Y(\nabla{}L)^TY -``` - -Note the property ``Y^T\mathtt{rgrad}(Y, \nabla{}L)\in\mathcal{S}_\mathrm{skew}(n).`` - -# Examples - -```jldoctest -using GeometricMachineLearning - -Y = StiefelManifold([1 0 ; 0 1 ; 0 0; 0 0]) -Δ = [1 2; 3 4; 5 6; 7 8] -rgrad(Y, Δ) - -# output - -4×2 Matrix{Int64}: - 0 -1 - 1 0 - 5 6 - 7 8 -``` -""" -function rgrad(Y::StiefelManifold, ∇L::AbstractMatrix) - ∇L - Y.A * (∇L' * Y.A) -end - -@doc raw""" - metric(Y::StiefelManifold, Δ₁::AbstractMatrix, Δ₂::AbstractMatrix) - -Compute the dot product for `Δ₁` and `Δ₂` at `Y`. - -This uses the canonical Riemannian metric for the Stiefel manifold: -```math -g_Y: (\Delta_1, \Delta_2) \mapsto \mathrm{Tr}(\Delta_1^T(\mathbb{I} - \frac{1}{2}YY^T)\Delta_2). -``` -""" -function metric(Y::StiefelManifold, Δ₁::AbstractMatrix, Δ₂::AbstractMatrix) - LinearAlgebra.tr(Δ₁'*(I - .5*Y.A*Y.A')*Δ₂) -end - -function check(Y::StiefelManifold) - norm(Y.A'*Y.A - I) -end - -@doc raw""" - global_section(Y::StiefelManifold) - -Compute a matrix of size ``N\times(N-n)`` whose columns are orthogonal to the columns in `Y`. - -This matrix is also called ``Y_\perp`` [absil2004riemannian, absil2008optimization, bendokat2020grassmann](@cite). - -# Examples - -```jldoctest -using GeometricMachineLearning -using GeometricMachineLearning: global_section -import Random - -Random.seed!(123) - -Y = StiefelManifold([1. 0.; 0. 1.; 0. 0.; 0. 0.]) - -round.(Matrix(global_section(Y)); digits = 3) - -# output - -4×2 Matrix{Float64}: - 0.0 -0.0 - 0.0 0.0 - 0.936 -0.353 - 0.353 0.936 -``` - -Further note that we convert the `QRCompactWYQ` object to a `Matrix` before we display it. - -# Implementation - -The implementation is done with a QR decomposition (`LinearAlgebra.qr!`). Internally we do: - -```julia -A = randn(N, N - n) # or the gpu equivalent -A = A - Y.A * (Y.A' * A) -qr!(A).Q -``` -""" -function global_section(Y::StiefelManifold{T}) where T - N, n = size(Y) - backend = networkbackend(Y) - A = KernelAbstractions.allocate(backend, T, N, N-n) - randn!(A) - A = A - Y.A * (Y.A' * A) - typeof(Y.A)(qr!(A).Q) -end - -GeometricOptimizers.global_section(Y::StiefelManifold) = global_section(Y) - -function GeometricOptimizers.apply_section!( - Y::StiefelManifold{T}, - λY::GeometricOptimizers.GlobalSection{T, <:StiefelManifold{T}}, - Y₂::StiefelManifold{T} -) where T - N, n = size(λY.Y) - @views Y.A .= λY.Y.A * Y₂.A[1:n, :] .+ λY.λ * Y₂.A[(n + 1):N, :] - Y -end - -@doc raw""" - Ω(Y::StiefelManifold{T}, Δ::AbstractMatrix{T}) where T - -Perform *canonical horizontal lift* for the Stiefel manifold: - -```math - \Delta \mapsto (\mathbb{I} - \frac{1}{2}YY^T)\Delta{}Y^T - Y\Delta^T(\mathbb{I} - \frac{1}{2}YY^T). -``` - -Internally this performs - -```julia -SkewSymMatrix(2 * (I(n) - .5 * Y * Y') * Δ * Y') -``` - -It uses [`SkewSymMatrix`](@ref) to save memory. - -# Examples - -```jldoctest -using GeometricMachineLearning -E = StiefelManifold(StiefelProjection(5, 2)) -Δ = [0. -1.; 1. 0.; 2. 3.; 4. 5.; 6. 7.] -GeometricMachineLearning.Ω(E, Δ) - -# output - -5×5 SkewSymMatrix{Float64, Vector{Float64}}: - 0.0 -1.0 -2.0 -4.0 -6.0 - 1.0 0.0 -3.0 -5.0 -7.0 - 2.0 3.0 0.0 -0.0 -0.0 - 4.0 5.0 0.0 0.0 -0.0 - 6.0 7.0 0.0 0.0 0.0 -``` - -Note that the output of `Ω` is a skew-symmetric matrix, i.e. an element of ``\mathfrak{g}``. -""" -function Ω(Y::StiefelManifold{T}, Δ::AbstractMatrix{T}) where T - YY = Y * Y' - SkewSymMatrix(2 * (one(YY) - T(.5) * Y * Y') * Δ * Y') -end - -function Base.copyto!(A::StiefelManifold, B::StiefelManifold) - A.A .= B.A - nothing -end - -Base.copy(A::StiefelManifold) = StiefelManifold(copy(A.A)) -Base.similar(A::StiefelManifold) = StiefelManifold(similar(A.A)) - -function Base.zero(Y::StiefelManifold{T}) where T - N, n = size(Y) - backend = networkbackend(Y.A) - zeros(backend, StiefelLieAlgHorMatrix{T}, N, n) -end - -# Bridge GML's StiefelManifold into GO's manifold optimization infrastructure. -# GO's methods dispatch on MT<:GO.Manifold{T}, but GML.StiefelManifold<:GML.Manifold{T} -# (a different type hierarchy), so we extend GO's functions explicitly for GML's type. - -function GeometricOptimizers.global_rep( - λY::GeometricOptimizers.GlobalSection{T, <:StiefelManifold{T}}, - Δ::AbstractMatrix{T} -) where T - N, n = size(λY.Y) - StiefelLieAlgHorMatrix( - SkewSymMatrix(λY.Y.A' * Δ), - λY.λ' * Δ, - N, n - ) -end - -@doc raw""" - geodesic(Y::StiefelManifold, Δ) - -Take as input an element `Y` of the [`StiefelManifold`](@ref) and a tangent vector `Δ` in the -corresponding tangent space and compute the geodesic (exponential map). - -In different notation: take as input an element ``x`` of ``\mathcal{M}`` and an element of -``T_x\mathcal{M}`` and return ``\mathtt{geodesic}(x, v_x) = \exp(v_x).`` - -# Examples - -```jldoctest -using GeometricMachineLearning - -Y = StiefelManifold([1. 0. 0.;]' |> Matrix) -Δ = [0. .5 0.;]' |> Matrix -Y₂ = geodesic(Y, Δ) - -Y₂' * Y₂ ≈ [1.;] - -# output - -true -``` - -# Implementation - -Internally this calls [`geodesic(::StiefelLieAlgHorMatrix)`](@ref). -""" -function GeometricOptimizers.geodesic(Y::StiefelManifold{T}, Δ::AbstractMatrix{T}) where T - λY = GeometricOptimizers.GlobalSection(Y) - B = GeometricOptimizers.global_rep(λY, Δ) - E = StiefelProjection(B) - expB = GeometricOptimizers.geodesic(B) - GeometricOptimizers.apply_section(λY, StiefelManifold(expB * E)) -end - -@doc raw""" - cayley(Y::StiefelManifold, Δ) - -Take as input an element `Y` of the [`StiefelManifold`](@ref) and a tangent vector `Δ` in the -corresponding tangent space and compute the Cayley retraction. - -In different notation: take as input an element ``x`` of ``\mathcal{M}`` and an element of -``T_x\mathcal{M}`` and return ``\mathrm{Cayley}(v_x).`` - -# Examples - -```jldoctest -using GeometricMachineLearning - -Y = StiefelManifold([1. 0. 0.;]' |> Matrix) -Δ = [0. .5 0.;]' |> Matrix -Y₂ = cayley(Y, Δ) - -Y₂' * Y₂ ≈ [1.;] - -# output - -true -``` - -# Implementation - -Internally this calls [`cayley(::StiefelLieAlgHorMatrix)`](@ref). -""" -function GeometricOptimizers.cayley(Y::StiefelManifold{T}, Δ::AbstractMatrix{T}) where T - λY = GeometricOptimizers.GlobalSection(Y) - B = GeometricOptimizers.global_rep(λY, Δ) - E = StiefelProjection(B) - cayleyB = GeometricOptimizers.cayley(B) - GeometricOptimizers.apply_section(λY, StiefelManifold(cayleyB * E)) -end - -function GeometricOptimizers.update_section!( - Λᵗ::GeometricOptimizers.GlobalSection{T, <:StiefelManifold{T}}, - Λ⁽ᵗ⁻¹⁾::GeometricOptimizers.GlobalSection{T, <:StiefelManifold{T}}, - B⁽ᵗ⁻¹⁾::AbstractMatrix{T}, - retraction -) where T - N, n = B⁽ᵗ⁻¹⁾.N, B⁽ᵗ⁻¹⁾.n - expB = retraction(B⁽ᵗ⁻¹⁾) - GeometricOptimizers.apply_section!(expB, Λ⁽ᵗ⁻¹⁾, expB) - Λᵗ.Y.A .= @view expB.A[:, 1:n] - Λᵗ.λ .= @view expB.A[:, (n+1):N] - nothing -end - -@doc raw""" - cayley(B̄::StiefelLieAlgHorMatrix) - -Compute the Cayley retraction of an element of [`StiefelLieAlgHorMatrix`](@ref). - -# Implementation - -We use the decomposition - -```math -\bar{B} = \begin{bmatrix} - A & -B^T \\ - B & \mathbb{O} -\end{bmatrix} = \begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix} =: B'(B'')^T -``` - -together with the Sherman-Morrison-Woodbury formula, so that only matrices of size -``2n\times2n`` have to be inverted. -""" -function GeometricOptimizers.cayley(B::StiefelLieAlgHorMatrix) - T = eltype(B) - E = StiefelProjection(B) - 𝕀_small = one(B.A) - 𝕆 = zero(𝕀_small) - 𝕀_small2 = hcat(vcat(𝕀_small, 𝕆), vcat(𝕆, 𝕀_small)) - 𝕀_big = one(B) - A_mat = B.A * 𝕀_small - B̂ = hcat(vcat(T(0.5) * A_mat, B.B), E) - B̄ = hcat(vcat(𝕀_small, T(0.5) * A_mat), vcat(zero(B.B'), -B.B'))' - StiefelManifold((𝕀_big + T(0.5) * B̂ * inv(𝕀_small2 - T(0.5) * B̄' * B̂) * B̄') * (𝕀_big + T(0.5) * B)) -end - -@doc raw""" - geodesic(B̄::StiefelLieAlgHorMatrix) - -Compute the geodesic of an element of [`StiefelLieAlgHorMatrix`](@ref). - -# Implementation - -Internally this is using - -```math -\mathbb{I} + B'\mathfrak{A}(B', B'')B'', -``` - -with the decomposition ``\bar{B} = B'(B'')^T`` described for -[`cayley(::StiefelLieAlgHorMatrix)`](@ref). ``\mathfrak{A}`` is a computationally efficient version -of the matrix exponential; it lives in `GeometricOptimizers` as `GeometricOptimizers.𝔄`. -""" -function GeometricOptimizers.geodesic(B::StiefelLieAlgHorMatrix) - T = eltype(B) - E = StiefelProjection(B) - unit = one(B.A) - A_mat = B.A * unit - B̂ = hcat(vcat(T(0.5) * A_mat, B.B), E) - B̄ = hcat(vcat(unit, T(0.5) * A_mat), vcat(zero(B.B'), -B.B'))' - StiefelManifold(one(B) + B̂ * GeometricOptimizers.𝔄(B̂, B̄) * B̄') -end - -function GeometricOptimizers._copyto!( - Λ₁::GeometricOptimizers.GlobalSection{T, <:StiefelManifold{T}}, - Λ₂::GeometricOptimizers.GlobalSection{T, <:StiefelManifold{T}} -) where T - copyto!(Λ₁.Y, Λ₂.Y) - copyto!(Λ₁.λ, Λ₂.λ) - Λ₁ -end - -# The elementwise arithmetic GO needs on `SkewSymMatrix` and `StiefelLieAlgHorMatrix` lives in -# `src/optimizers/go_bridges.jl`, together with the same bridges for the other GML array types. diff --git a/src/optimizers/go_bridges.jl b/src/optimizers/go_bridges.jl deleted file mode 100644 index 83f5127e7..000000000 --- a/src/optimizers/go_bridges.jl +++ /dev/null @@ -1,108 +0,0 @@ -# Elementwise arithmetic bridges between GML's structured matrix types and GeometricOptimizers. -# -# GeometricOptimizers builds its optimizer caches out of a handful of in-place primitives -- `_add!`, -# `_rac!` (elementwise square root), `_square!`, `_div!` and `_rmul!`. Its generic methods broadcast -# over `AbstractArray`, which does not work for GML's structured matrices: they store only their free -# parameters in a vector `S` (or `A`/`B`) and either have no `setindex!` at all or would silently -# symmetrise what is written to them. -# -# For every one of these types the free parameters *are* the coordinates the optimizer should work -# in, so each bridge is the corresponding operation on the storage. GO dispatches on its own -# `SkewSymMatrix`/`StiefelLieAlgHorMatrix`, which are distinct types from GML's, hence the need to -# define these here rather than relying on GO's own methods. -# -# `GeometricOptimizers.update_section!` is bridged for the same reason: its Euclidean method is -# `Λᵗ.Y .= Λ⁽ᵗ⁻¹⁾.Y .+ B⁽ᵗ⁻¹⁾`, a broadcast into the parameter. - -# --- SkewSymMatrix --------------------------------------------------------------------------- - -GeometricOptimizers._add!(a::SkewSymMatrix{T}, b::SkewSymMatrix{T}) where T = (a.S .+= b.S; a) -GeometricOptimizers._add!(a::SkewSymMatrix{T}, b::T) where T = (a.S .+= b; a) -GeometricOptimizers._rac!(B::SkewSymMatrix, A::SkewSymMatrix) = (B.S .= sqrt.(A.S); B) -GeometricOptimizers._square!(B::SkewSymMatrix, A::SkewSymMatrix) = (B.S .= A.S .^ 2; B) -function GeometricOptimizers._div!(C::SkewSymMatrix, A::SkewSymMatrix, B::SkewSymMatrix) - C.S .= A.S ./ B.S - C -end - -# --- StiefelLieAlgHorMatrix ------------------------------------------------------------------ - -function GeometricOptimizers._add!(A::StiefelLieAlgHorMatrix{T}, - B::StiefelLieAlgHorMatrix{T}) where T - GeometricOptimizers._add!(A.A, B.A) - A.B .+= B.B - A -end -function GeometricOptimizers._add!(A::StiefelLieAlgHorMatrix{T}, b::T) where T - GeometricOptimizers._add!(A.A, b) - A.B .+= b - A -end -function GeometricOptimizers._rac!(B::StiefelLieAlgHorMatrix, A::StiefelLieAlgHorMatrix) - GeometricOptimizers._rac!(B.A, A.A) - B.B .= sqrt.(A.B) - B -end -function GeometricOptimizers._square!(B::StiefelLieAlgHorMatrix, A::StiefelLieAlgHorMatrix) - GeometricOptimizers._square!(B.A, A.A) - B.B .= A.B .^ 2 - B -end -function GeometricOptimizers._div!(C::StiefelLieAlgHorMatrix, A::StiefelLieAlgHorMatrix, - B::StiefelLieAlgHorMatrix) - GeometricOptimizers._div!(C.A, A.A, B.A) - C.B .= A.B ./ B.B - C -end - -# --- GrassmannLieAlgHorMatrix ---------------------------------------------------------------- - -function GeometricOptimizers._add!(A::GrassmannLieAlgHorMatrix, B::GrassmannLieAlgHorMatrix) - A.B .+= B.B - A -end -GeometricOptimizers._add!(A::GrassmannLieAlgHorMatrix, b::Number) = (A.B .+= b; A) -function GeometricOptimizers._rac!(B::GrassmannLieAlgHorMatrix, A::GrassmannLieAlgHorMatrix) - B.B .= sqrt.(A.B) - B -end -function GeometricOptimizers._square!(B::GrassmannLieAlgHorMatrix, A::GrassmannLieAlgHorMatrix) - B.B .= A.B .^ 2 - B -end -function GeometricOptimizers._div!(C::GrassmannLieAlgHorMatrix, A::GrassmannLieAlgHorMatrix, - B::GrassmannLieAlgHorMatrix) - C.B .= A.B ./ B.B - C -end - -# --- Euclidean parameters with structured storage --------------------------------------------- -# -# These are ordinary vector-space parameters (`SymmetricMatrix` in the SympNet and symplectic -# attention layers, `SkewSymMatrix` in volume-preserving attention, `LowerTriangular` and -# `UpperTriangular` in the volume-preserving feedforward layers), so the optimizer treats them like -# any other array -- it just cannot broadcast into them. - -for MT in (:SymmetricMatrix, :LowerTriangular, :UpperTriangular) - @eval begin - GeometricOptimizers._add!(a::$MT{T}, b::$MT{T}) where T = (a.S .+= b.S; a) - GeometricOptimizers._add!(a::$MT{T}, b::T) where T = (a.S .+= b; a) - GeometricOptimizers._rac!(B::$MT, A::$MT) = (B.S .= sqrt.(A.S); B) - GeometricOptimizers._square!(B::$MT, A::$MT) = (B.S .= A.S .^ 2; B) - GeometricOptimizers._div!(C::$MT, A::$MT, B::$MT) = (C.S .= A.S ./ B.S; C) - end -end - -for MT in (:SymmetricMatrix, :SkewSymMatrix, :LowerTriangular, :UpperTriangular) - @eval begin - GeometricOptimizers._rmul!(a::$MT, b) = (rmul!(a.S, b); a) - - function GeometricOptimizers.update_section!( - Λᵗ::GeometricOptimizers.GlobalSection{T, <:$MT{T}, Nothing}, - Λ⁽ᵗ⁻¹⁾::GeometricOptimizers.GlobalSection{T, <:$MT{T}, Nothing}, - B⁽ᵗ⁻¹⁾::$MT{T}, retraction) where T - Λᵗ.Y.S .= Λ⁽ᵗ⁻¹⁾.Y.S .+ B⁽ᵗ⁻¹⁾.S - Λᵗ - end - end -end diff --git a/src/optimizers/optimizer.jl b/src/optimizers/optimizer.jl index 94ec9b944..e26831cca 100644 --- a/src/optimizers/optimizer.jl +++ b/src/optimizers/optimizer.jl @@ -31,30 +31,22 @@ end GMLEuclideanState(x::AbstractArray{T}) where T = GMLEuclideanState{T, typeof(x)}(0, zero(x), zero(x)) -"""Adam optimizer method with exponential learning-rate decay.""" -struct AdamOptimizerWithDecay{T<:Real} <: GeometricOptimizers.OptimizerMethod - η₁::T; η₂::T; ρ₁::T; ρ₂::T; δ::T; γ::T; n_epochs::Int - function AdamOptimizerWithDecay(n_epochs::Int, η₁=1f-2, η₂=1f-6, - ρ₁=9f-1, ρ₂=9.9f-1, δ=1f-8; T=typeof(η₁)) - γ = exp(log(η₂/η₁) / n_epochs) - new{T}(T(η₁), T(η₂), T(ρ₁), T(ρ₂), T(δ), T(γ), n_epochs) - end -end +# `AdamOptimizerWithDecay` used to be defined here, as an `OptimizerMethod` bundling Adam's `ρ₁`, +# `ρ₂`, `δ` with a learning-rate schedule `η₁`, `η₂`, `n_epochs`. GeometricOptimizers ships the same +# algorithm — the same `γ = exp(log(η₂/η₁)/n)` — split the way it belongs: the direction is an +# `Adam` method, the schedule is a `DecayingStatic` line search. Both names are imported, and +# `Optimizer` below takes a `DecayingStatic` as its `step_size`. Two packages exporting the name was +# issue B1: `using GeometricMachineLearning, GeometricOptimizers` failed outright on it. _is_go_native_method(::GeometricOptimizers.GradientMethod) = true _is_go_native_method(::GeometricOptimizers.MomentumMethod) = true _is_go_native_method(::GeometricOptimizers.Adam) = true -# `AdamOptimizerWithDecay` differs from `Adam` only in the step size, which GML supplies separately -# through `_current_step_size`, so it uses GO's Adam cache and state like any other Adam. -_is_go_native_method(::AdamOptimizerWithDecay) = true _is_go_native_method(::GeometricOptimizers.OptimizerMethod) = false _adapt_method_to_T(method::GeometricOptimizers.Adam, ::Type{T}) where T = GeometricOptimizers.Adam(T; β₁ = T(method.β₁), β₂ = T(method.β₂), δ = T(method.δ)) _adapt_method_to_T(method::GeometricOptimizers.MomentumMethod, ::Type{T}) where T = GeometricOptimizers.MomentumMethod(T(method.α)) -_adapt_method_to_T(method::AdamOptimizerWithDecay, ::Type{T}) where T = - GeometricOptimizers.Adam(T; β₁ = T(method.ρ₁), β₂ = T(method.ρ₂), δ = T(method.δ)) _adapt_method_to_T(method, ::Type) = method _use_go_cache(method, x) = @@ -80,40 +72,86 @@ function _make_optimizer_state(method, x) end end -"""Optimizer state combining a GeometricOptimizers method with GML parameters.""" -mutable struct Optimizer{MT <: GeometricOptimizers.OptimizerMethod, CT, ST, RT} +""" + Optimizer(method, nn; retraction, step_size) + Optimizer(nn; algorithm, linesearch, retraction) + +Optimizer state combining a `GeometricOptimizers` method with the parameters of a neural network. + +`step_size` is either a number — a fixed learning rate — or a +`GeometricOptimizers.DecayingStatic`, a learning rate that decays geometrically with the iteration +number. The second form is the one `GeometricOptimizers` uses itself, so a method paired with a +schedule splats straight in: + +```julia +opt = Optimizer(nn; AdamOptimizerWithDecay(n_epochs, Float32)...) +``` + +# Extended help + +The step size is a property of the optimizer and not of the method: the same `Adam()` trains at any +learning rate. That is the split `GeometricOptimizers` makes — the method supplies a direction, a +`SimpleSolvers.LinesearchMethod` supplies how far to go along it — and `step_size` is GML's half of +it for a training loop, which has no objective function for a real line search to evaluate. +""" +mutable struct Optimizer{MT <: GeometricOptimizers.OptimizerMethod, CT, ST, RT, SST} method::MT cache::CT state::ST retraction::RT - step_size::Float64 + step_size::SST iterations::Int end _default_step_size(::GeometricOptimizers.Adam) = 1e-3 -_default_step_size(method::AdamOptimizerWithDecay) = Float64(method.η₁) _default_step_size(::GeometricOptimizers.OptimizerMethod) = 1e-2 -_current_step_size(opt::Optimizer, ::Int) = opt.step_size -_current_step_size(opt::Optimizer{<:AdamOptimizerWithDecay}, t::Int) = - Float64(opt.method.η₁ * opt.method.γ^t) +_step_size(η::Real, ::Int) = Float64(η) +# `t` and not `t - 1`: `optimization_step!` increments before it asks, so the first step of a solve +# is `α(1) = γη₁`. That is what `DecayingStatic` means by iteration `t` — `solve!` calls +# `increase_iteration_number!` before `solver_step!` — and it is what GML's own +# `AdamOptimizerWithDecay` did before the schedule moved upstream. +_step_size(ls::DecayingStatic, t::Int) = Float64(GeometricOptimizers.step_size(ls, t)) + +_current_step_size(opt::Optimizer, t::Int) = _step_size(opt.step_size, t) + +_optimizer_step_size(η::Real) = Float64(η) +_optimizer_step_size(ls::DecayingStatic) = ls function Optimizer(method::GeometricOptimizers.OptimizerMethod, nn::NeuralNetwork; retraction = GeometricOptimizers.cayley, - step_size::Real = _default_step_size(method)) - ps = params(nn) - Optimizer(method, _make_optimizer_cache(method, ps), _make_optimizer_state(method, ps), - retraction, Float64(step_size), 0) + step_size = _default_step_size(method)) + Optimizer(method, params(nn); retraction = retraction, step_size = step_size) end function Optimizer(method::GeometricOptimizers.OptimizerMethod, ps::Union{NamedTuple, NeuralNetworkParameters}; retraction = GeometricOptimizers.cayley, - step_size::Real = _default_step_size(method)) + step_size = _default_step_size(method)) Optimizer(method, _make_optimizer_cache(method, ps), _make_optimizer_state(method, ps), - retraction, Float64(step_size), 0) + retraction, _optimizer_step_size(step_size), 0) end +# The keyword form, so that the `(algorithm, linesearch)` pairing `GeometricOptimizers` returns from +# `AdamOptimizerWithDecay` splats in unchanged. `linesearch` is the step size under the name +# upstream gives it; a `Static` carries its own `α`, which is then the fixed learning rate. +function Optimizer(nn_or_ps::Union{NeuralNetwork, NamedTuple, NeuralNetworkParameters}; + algorithm::GeometricOptimizers.OptimizerMethod, + linesearch = nothing, + retraction = GeometricOptimizers.cayley, + step_size = linesearch === nothing ? _default_step_size(algorithm) : + _step_size_from_linesearch(linesearch)) + Optimizer(algorithm, nn_or_ps; retraction = retraction, step_size = step_size) +end + +_step_size_from_linesearch(ls::DecayingStatic) = ls +_step_size_from_linesearch(ls::GeometricOptimizers.Static) = Float64(ls.α) +_step_size_from_linesearch(ls) = throw(ArgumentError( + "`Optimizer` takes a fixed step size or a `DecayingStatic` schedule, not a $(typeof(ls)). " * + "A training loop evaluates its loss on one batch at a time and has no objective for a line " * + "search to search along; use `GeometricOptimizers.Optimizer` with an `OptimizerProblem` for " * + "that.")) + # Euclidean update rules function _euclidean_update!(x::AbstractArray{T}, dx::AbstractArray, state::GMLEuclideanState, ::GeometricOptimizers.GradientMethod, step_size) where T @@ -139,17 +177,9 @@ function _euclidean_update!(x::AbstractArray{T}, dx::AbstractArray, state.m₂ .= fac₂₁ .* state.m₂ .+ fac₂₂ .* dx .^ 2 x .-= T(step_size) .* state.m₁ ./ (sqrt.(state.m₂) .+ δ) end -function _euclidean_update!(x::AbstractArray{T}, dx::AbstractArray, - state::GMLEuclideanState{T}, method::AdamOptimizerWithDecay, step_size) where T - t = state.iterations; _t = t + 1 - ρ₁, ρ₂, δ = T(method.ρ₁), T(method.ρ₂), T(method.δ) - # see the note in the `Adam` method above - fac₁₁ = (ρ₁-ρ₁^_t)/(1-ρ₁^_t); fac₁₂ = (1-ρ₁)/(1-ρ₁^_t) - fac₂₁ = (ρ₂-ρ₂^_t)/(1-ρ₂^_t); fac₂₂ = (1-ρ₂)/(1-ρ₂^_t) - state.m₁ .= fac₁₁ .* state.m₁ .+ fac₁₂ .* dx - state.m₂ .= fac₂₁ .* state.m₂ .+ fac₂₂ .* dx .^ 2 - x .-= T(step_size) .* state.m₁ ./ (sqrt.(state.m₂) .+ δ) -end +# There used to be a fourth method here, for `AdamOptimizerWithDecay`, character for character the +# `Adam` one above with `ρ₁`, `ρ₂` in place of `β₁`, `β₂`. Adam with a decaying learning rate *is* +# Adam — only `step_size` differs, and that comes in as an argument — so the `Adam` method serves it. function _go_update_leaf!(cache, state, local_grad, method::GeometricOptimizers.Adam, ps_leaf) @@ -233,17 +263,26 @@ end """ optimization_step!(opt, λY, ps, dp) -Apply one optimization step to the parameters `ps` and their gradient `dp`. +Apply one optimization step to the parameters `ps` and their gradient `dp`, with the method and step +size the [`Optimizer`](@ref) `opt` carries. `λY` is a `GlobalSection` of `ps` (or a `NamedTuple` of them). Note that it is an *output* here: the section the optimizer carries from step to step lives in `opt.state`, and `λY` is written so that callers who inspect it see the updated section. It therefore has to be allocated once and reused, not rebuilt per step -- rebuilding it costs a QR decomposition per manifold weight. + +The step counter is incremented *before* the step size is read, so the first step of a run is step 1. +This matters for a decaying `step_size` and is how `GeometricOptimizers` counts too. """ function optimization_step!(opt::Optimizer, λY, ps, dp) + # The increment comes *first*, so the first step of a run is step 1. It matters only for a + # decaying `step_size`, and there it matters: reading the schedule before incrementing takes + # `α(0) = η₁`, one whole step above what the pre-0.5 `AdamOptimizerWithDecay` took and what + # `DecayingStatic` and `GeometricOptimizers.solve!` take. `solve!` counts the same way, by + # calling `increase_iteration_number!` before `solver_step!`. + opt.iterations += 1 step = _current_step_size(opt, opt.iterations) _tree_optim_step!(opt.cache, opt.state, dp, ps, λY, opt.method, opt.retraction, step) - opt.iterations += 1 nothing end diff --git a/src/utils.jl b/src/utils.jl index 7ec85b98c..9fa719360 100644 --- a/src/utils.jl +++ b/src/utils.jl @@ -106,10 +106,8 @@ function center_align_text(text, width) return aligned_text end -# The following are fallback functions - maybe you want to put them into a separate file -function global_section(::AbstractVecOrMat) - nothing -end +# `global_section(::AbstractVecOrMat) = nothing` used to be defined here, identically to +# GeometricOptimizers' own fallback. It is imported now. """ QPT From 6e334df270c5930d3efac31f2f2e0074e9dfe4f1 Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Mon, 17 Aug 2026 21:58:30 +0900 Subject: [PATCH 02/12] Delete the tests that were GeometricOptimizers', quiet the rest Twenty-three files go. They fall into three groups. **Twelve duplicated upstream's suite.** `test/arrays/` and `test/manifolds/` tested the eleven types the previous commit stopped defining, so they were a second copy of `test/special_matrices/`, `test/lie_algebras/`, `test/manifolds/` and `test/global_sections/` over there -- against the same types, now that the types are the same. Each was diffed against its upstream counterpart first and what upstream did not already assert was ported there (GeometricOptimizers#50); `test/arrays/poisson_tensor.jl` stays, because `PoissonTensor` is GML's. Folding them in found four defects in upstream's suite, including three functions defined and never called and a `stiefel_global_section` that built a `GrassmannManifold` -- so the Stiefel global section had no test at all. Fixed there. **Eight were unreachable from `runtests.jl` and could not have run.** `optimizers/lie_alg_lifts.jl` and `optimizers/hor_lift.jl` include `../src/arrays/skew_sym.jl` and `../src/optimizers/householder.jl`, paths that stopped existing long before this branch. Others `using Lux`, which is not a dependency. Nothing ran them, so nothing reported the rot. **Three tested behaviour that is now upstream's**, including `test/docstrings/manifolds.jl`, whose docstrings moved with the documentation. ## What stayed, and why `optimizer_convergence_tests/{svd,psd}_optim.jl`, `optimizers/{gradient,momentum}_optimizer.jl`, `optimizers/structured_array_parameters.jl`, `optimizers/utils/optimization_step.jl`, `layers/manifold_layers.jl` and `transformer_related/*` all drive `Chain`/`NeuralNetwork`/`optimization_step!`. They look like upstream's `test/optimizer_convergence/svd_optim.jl` and are not: that one drives `solve!` against an objective, these train a network with gradients from Zygote. Different code path, both worth having. ## New coverage `adam_with_learning_rate_decay.jl` gains two tests for the previous commit's fixes: that the pairing works on `StiefelManifold` weights (GML's own `AdamOptimizerWithDecay` was a distinct `OptimizerMethod` that had to be routed onto Adam's cache explicitly, and without the routing manifold weights fell through to the Euclidean state, whose zero element is not a manifold point), and that the schedule is walked from `t = 1`, asserted both through `_current_step_size` and through `optimization_step!`. ## Progress bars A test that trains passes `show_progress = false` now -- nine call sites. The `Optimizer` functor defaults it to `true`, which is right at a REPL and is noise in a suite: the 2048-epoch run in `adam_with_learning_rate_decay.jl` alone emitted a few hundred progress lines, which is what a failure would have had to be found in. The default is unchanged, so interactive use is unaffected, and `runtests.jl` records the convention so it does not creep back. Co-Authored-By: Claude Opus 5 (1M context) --- .../addition_tests_for_custom_arrays.jl | 44 ------- test/arrays/array_tests.jl | 112 ------------------ .../constructor_tests_for_custom_arrays.jl | 49 -------- test/arrays/map_to_skew.jl | 8 -- ...matrix_multiplication_for_custom_arrays.jl | 24 ---- .../random_generation_of_custom_arrays.jl | 29 ----- ...scalar_multiplication_for_custom_arrays.jl | 43 ------- test/arrays/skew_sym_conv_test.jl | 23 ---- test/arrays/symmetric_matrix.jl | 29 ----- ...test_grassmann_lie_alg_hor_constructors.jl | 47 -------- .../test_stiefel_lie_alg_hor_constructors.jl | 48 -------- test/arrays/triangular.jl | 37 ++---- test/data_loader/batch_data_loader_qp_test.jl | 4 +- .../data_loader_for_input_and_output.jl | 2 +- test/manifolds/grassmann_manifold.jl | 73 ------------ test/manifolds/stiefel_manifold.jl | 65 ---------- test/manifolds/symplectic_stiefel_manifold.jl | 30 ----- .../network_losses/losses_and_optimization.jl | 2 +- test/optimizers/exponential_retractions.jl | 41 ------- test/optimizers/hor_lift.jl | 4 - test/optimizers/lie_alg_lifts.jl | 21 ---- test/optimizers/manifold_optim.jl | 41 ------- .../manifold_related/legacy_functions.jl | 27 ----- test/optimizers/momentum_optim_test.jl | 29 ----- .../adam_with_learning_rate_decay.jl | 54 +++++++-- test/optimizers/riemannian_gradients.jl | 36 ------ test/optimizers/standard_optim_test.jl | 34 ------ .../optimizers/structured_array_parameters.jl | 25 ++-- test/optimizers/utils/global_sections.jl | 37 ------ test/runtests.jl | 41 +------ test/sae_error_lower_than_psd_error.jl | 2 +- test/symplectic_autoencoder_tests.jl | 4 +- ...ulti_head_attention_stiefel_optim_cache.jl | 1 + ...multi_head_attention_stiefel_retraction.jl | 3 + .../multi_head_attention_stiefel_setup.jl | 2 + 35 files changed, 93 insertions(+), 978 deletions(-) delete mode 100644 test/arrays/addition_tests_for_custom_arrays.jl delete mode 100644 test/arrays/array_tests.jl delete mode 100644 test/arrays/constructor_tests_for_custom_arrays.jl delete mode 100644 test/arrays/map_to_skew.jl delete mode 100644 test/arrays/matrix_multiplication_for_custom_arrays.jl delete mode 100644 test/arrays/random_generation_of_custom_arrays.jl delete mode 100644 test/arrays/scalar_multiplication_for_custom_arrays.jl delete mode 100644 test/arrays/skew_sym_conv_test.jl delete mode 100644 test/arrays/symmetric_matrix.jl delete mode 100644 test/arrays/test_grassmann_lie_alg_hor_constructors.jl delete mode 100644 test/arrays/test_stiefel_lie_alg_hor_constructors.jl delete mode 100644 test/manifolds/grassmann_manifold.jl delete mode 100644 test/manifolds/stiefel_manifold.jl delete mode 100644 test/manifolds/symplectic_stiefel_manifold.jl delete mode 100644 test/optimizers/exponential_retractions.jl delete mode 100644 test/optimizers/hor_lift.jl delete mode 100644 test/optimizers/lie_alg_lifts.jl delete mode 100644 test/optimizers/manifold_optim.jl delete mode 100644 test/optimizers/manifold_related/legacy_functions.jl delete mode 100644 test/optimizers/momentum_optim_test.jl delete mode 100644 test/optimizers/riemannian_gradients.jl delete mode 100644 test/optimizers/standard_optim_test.jl delete mode 100644 test/optimizers/utils/global_sections.jl diff --git a/test/arrays/addition_tests_for_custom_arrays.jl b/test/arrays/addition_tests_for_custom_arrays.jl deleted file mode 100644 index 55d96028f..000000000 --- a/test/arrays/addition_tests_for_custom_arrays.jl +++ /dev/null @@ -1,44 +0,0 @@ -using GeometricMachineLearning, Test -import Random - -Random.seed!(1234) - -@doc raw""" -This function tests addition for various custom arrays, i.e. if \(A + B\) is performed in the correct way. -""" -function addition_tests_for_custom_arrays(n::Int, N::Int, T::Type) - A = rand(T, n, n) - B = rand(T, n, n) - - # SymmetricMatrix - AB_sym = SymmetricMatrix(A + B) - AB_sym2 = SymmetricMatrix(A) + SymmetricMatrix(B) - @test AB_sym ≈ AB_sym2 - @test typeof(AB_sym) <: SymmetricMatrix{T} - @test typeof(AB_sym2) <: SymmetricMatrix{T} - - # SkewSymMatrix - AB_skew = SkewSymMatrix(A + B) - AB_skew2 = SkewSymMatrix(A) + SkewSymMatrix(B) - @test AB_skew ≈ AB_skew2 - @test typeof(AB_skew) <: SkewSymMatrix{T} - @test typeof(AB_skew2) <: SkewSymMatrix{T} - - C = rand(T, N, N) - D = rand(T, N, N) - - # StiefelLieAlgHorMatrix - CD_slahm = StiefelLieAlgHorMatrix(C + D, n) - CD_slahm2 = StiefelLieAlgHorMatrix(C, n) + StiefelLieAlgHorMatrix(D, n) - @test CD_slahm ≈ CD_slahm2 - @test typeof(CD_slahm) <: StiefelLieAlgHorMatrix{T} - @test typeof(CD_slahm2) <: StiefelLieAlgHorMatrix{T} - - CD_glahm = GrassmannLieAlgHorMatrix(C + D, n) - CD_glahm2 = GrassmannLieAlgHorMatrix(C, n) + GrassmannLieAlgHorMatrix(D, n) - @test CD_glahm ≈ CD_glahm2 - @test typeof(CD_glahm) <: GrassmannLieAlgHorMatrix{T} - @test typeof(CD_glahm2) <: GrassmannLieAlgHorMatrix{T} -end - -addition_tests_for_custom_arrays(5, 10, Float32) \ No newline at end of file diff --git a/test/arrays/array_tests.jl b/test/arrays/array_tests.jl deleted file mode 100644 index f2baca095..000000000 --- a/test/arrays/array_tests.jl +++ /dev/null @@ -1,112 +0,0 @@ -using LinearAlgebra -using Random -using Test -using GeometricMachineLearning - -Random.seed!(1234) - -#check if symmetric matrix works for 1×1 matrices -W = rand(1,1) -S = SymmetricMatrix(W) -@test abs(W[1,1] - S[1,1]) < 1e-10 - -#check if built-in projection, matrix addition & subtraction works -function sym_mat_add_sub_test(n) - symmetrize(W) = .5*(W + W') - W₁ = rand(n,n) - S₁ = SymmetricMatrix(W₁) - W₂ = rand(n,n) - S₂ = SymmetricMatrix(W₂) - S₃ = S₁ + S₂ - S₄ = S₁ - S₂ - @test typeof(S₃) <: SymmetricMatrix - @test typeof(S₄) <: SymmetricMatrix - @test all(abs.(symmetrize(W₁ + W₂) .- S₃) .< 1e-10) - @test all(abs.(symmetrize(W₁ - W₂) .- S₄) .< 1e-10) -end - -function skew_mat_add_sub_test(n) - anti_symmetrize(W) = .5*(W - W') - W₁ = rand(n,n) - S₁ = SkewSymMatrix(W₁) - W₂ = rand(n,n) - S₂ = SkewSymMatrix(W₂) - S₃ = S₁ + S₂ - S₄ = S₁ - S₂ - @test typeof(S₃) <: SkewSymMatrix - @test typeof(S₄) <: SkewSymMatrix - @test all(abs.(anti_symmetrize(W₁ + W₂) .- S₃) .< 1e-10) - @test all(abs.(anti_symmetrize(W₁ - W₂) .- S₄) .< 1e-10) -end - -# this function tests if the matrix multiplication for the SkewSym Matrix is the same as the implied one. -function skew_mat_mul_test(n, T=Float64) - S = rand(SkewSymMatrix{T}, n) - A = rand(n, n) - SA1 = S*A - SA2 = Matrix{T}(S)*A - @test isapprox(SA1, SA2) -end - -function skew_mat_mul_test2(n, T=Float64) - S = rand(SkewSymMatrix{T}, n) - A = rand(n, n) - AS1 = A*S - AS2 = A*Matrix{T}(S) - @test isapprox(AS1, AS2) -end - -# test Stiefel manifold projection test -function stiefel_proj_test(N,n) - In = I(n) - E = StiefelProjection(N, n, Float64) - @test all(abs.((E'*E) .- In) .< 1e-10) -end - -function stiefel_lie_alg_add_sub_test(N, n) - E = StiefelProjection(N, n) - projection(W::SkewSymMatrix) = W - (I - E*E')*W*(I - E*E') - W₁ = SkewSymMatrix(rand(N,N)) - S₁ = StiefelLieAlgHorMatrix(W₁,n) - W₂ = SkewSymMatrix(rand(N,N)) - S₂ = StiefelLieAlgHorMatrix(W₂,n) - A = rand(N, N) - S₃ = S₁ + S₂ - S₄ = S₁ - S₂ - @test typeof(S₃) <: StiefelLieAlgHorMatrix - @test typeof(S₄) <: StiefelLieAlgHorMatrix - @test all(abs.(projection(W₁ + W₂) .- S₃) .< 1e-10) - @test all(abs.(projection(W₁ - W₂) .- S₄) .< 1e-10) - # check custom addition - @test S₁ + A ≈ Matrix(S₁) + A - @test A + S₁ ≈ Matrix(S₁) + A -end - - -function stiefel_lie_alg_vectorization_test(N, n; T=Float32) - A = rand(StiefelLieAlgHorMatrix{T}, N, n) - @test isapprox(StiefelLieAlgHorMatrix(vec(A), N, n), A) -end - -# TODO: tests for ADAM functions - -# test everything for different n & N values -Random.seed!(42) - -N_max = 20 -n_max = 10 -num = 100 - -N_vec = Int.(ceil.(rand(num)*N_max)) -n_vec = Int.(ceil.(rand(num)*n_max)) -n_vec = min.(n_vec, N_vec) - -for (N, n) ∈ zip(N_vec, n_vec) - sym_mat_add_sub_test(N) - skew_mat_add_sub_test(N) - skew_mat_mul_test(N) - skew_mat_mul_test2(N) - stiefel_proj_test(N,n) - stiefel_lie_alg_add_sub_test(N,n) - stiefel_lie_alg_vectorization_test(N, n) -end diff --git a/test/arrays/constructor_tests_for_custom_arrays.jl b/test/arrays/constructor_tests_for_custom_arrays.jl deleted file mode 100644 index c0d1a2c09..000000000 --- a/test/arrays/constructor_tests_for_custom_arrays.jl +++ /dev/null @@ -1,49 +0,0 @@ -using GeometricMachineLearning, Test -using LinearAlgebra: I -import Random - -Random.seed!(1234) - -@doc raw""" -This tests various constructor for custom arrays, e.g. if calling `SymmetricMatrix` on a matrix ``A`` does -```math -A \mapsto \frac{1}{2}(A + A^T). -``` -""" -function test_constructors_for_custom_arrays(n::Int, N::Int, T::Type) - A = rand(T, n, n) - B = rand(T, N, N) - - # SymmetricMatrix - @test Matrix{T}(SymmetricMatrix(A)) ≈ T(.5) * (A + A') - - # SkewSymMatrix - @test Matrix{T}(SkewSymMatrix(A)) ≈ T(.5) * (A - A') - - # StiefelLieAlgHorMatrix - B_shor = StiefelLieAlgHorMatrix(SkewSymMatrix(B), n) - B_shor2 = Matrix{T}(SkewSymMatrix(B)) - B_shor2[(n+1):N, (n+1):N] .= zero(T) - @test Matrix{T}(B_shor) ≈ B_shor2 - - # GrassmannLieAlgHorMatrix - B_ghor = GrassmannLieAlgHorMatrix(SkewSymMatrix(B), n) - B_ghor2 = copy(B_shor2) - B_ghor2[1:n, 1:n] .= zero(T) - @test Matrix{T}(B_ghor) ≈ B_ghor2 - - # StiefelProjection - E = StiefelProjection(T, N, n) - @test Matrix{T}(E) ≈ vcat(I(n), zeros(T, (N-n), n)) -end - -test_constructors_for_custom_arrays(5, 10, Float32) - -@test LowerTriangular([1, 2, 3, 4, 5, 6], 4) == - [0 0 0 0; 1 0 0 0; 2 3 0 0; 4 5 6 0] -@test UpperTriangular([1, 2, 3, 4, 5, 6], 4) == - [0 1 2 4; 0 0 3 5; 0 0 0 6; 0 0 0 0] -@test SymmetricMatrix([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 4) == - [1 2 4 7; 2 3 5 8; 4 5 6 9; 7 8 9 10] -@test SkewSymMatrix([1, 2, 3, 4, 5, 6], 4) == - [0 -1 -2 -4; 1 0 -3 -5; 2 3 0 -6; 4 5 6 0] diff --git a/test/arrays/map_to_skew.jl b/test/arrays/map_to_skew.jl deleted file mode 100644 index 58f5e9063..000000000 --- a/test/arrays/map_to_skew.jl +++ /dev/null @@ -1,8 +0,0 @@ -using GeometricMachineLearning: SkewSymMatrix, map_to_Skew - -function test_map_to_Skew(n::Int = 5) - A = rand(SkewSymMatrix, n) - @test A.S ≈ map_to_Skew(A) -end - -test_map_to_Skew() \ No newline at end of file diff --git a/test/arrays/matrix_multiplication_for_custom_arrays.jl b/test/arrays/matrix_multiplication_for_custom_arrays.jl deleted file mode 100644 index 6ac83e1e1..000000000 --- a/test/arrays/matrix_multiplication_for_custom_arrays.jl +++ /dev/null @@ -1,24 +0,0 @@ -using GeometricMachineLearning, Test -import Random - -Random.seed!(1234) - -@doc raw""" -This function tests matrix multiplication for various custom arrays, i.e. if \((A,\alpha) \mapsto \alpha{}A\) is performed in the correct way. -""" -function matrix_multiplication_tests_for_custom_arrays(n::Int, N::Int, T::Type) - A = rand(T, n, n) - B = rand(T, n, N) - - # SymmetricMatrix - A_sym = SymmetricMatrix(A) - @test A_sym * B ≈ Matrix{T}(A_sym) * B - @test B' * A_sym ≈ B' * Matrix{T}(A_sym) - - # SkewSymMatrix - A_skew = SkewSymMatrix(A) - @test A_skew * B ≈ Matrix{T}(A_skew) * B - @test B' * A_skew ≈ B' * Matrix{T}(A_skew) -end - -matrix_multiplication_tests_for_custom_arrays(5, 10, Float32) \ No newline at end of file diff --git a/test/arrays/random_generation_of_custom_arrays.jl b/test/arrays/random_generation_of_custom_arrays.jl deleted file mode 100644 index 97550e625..000000000 --- a/test/arrays/random_generation_of_custom_arrays.jl +++ /dev/null @@ -1,29 +0,0 @@ -using LinearAlgebra -using Random -using Test -using GeometricMachineLearning - -Random.seed!(1234) - -""" -This tests random generation of custom arrays. This will have to be expanded to GPU tests. -""" -function test_random_array_generation(n::Int, N::Int, T::Type) - A_sym = rand(SymmetricMatrix{T}, n) - @test typeof(A_sym) <: SymmetricMatrix{T} - @test eltype(A_sym) == T - - A_skew = rand(SkewSymMatrix{T}, n) - @test typeof(A_skew) <: SkewSymMatrix{T} - @test eltype(A_skew) == T - - A_stiefel_hor = rand(StiefelLieAlgHorMatrix{T}, N, n) - @test typeof(A_stiefel_hor) <: StiefelLieAlgHorMatrix{T} - @test eltype(A_stiefel_hor) == T - - A_grassmann_hor = rand(GrassmannLieAlgHorMatrix{T}, N, n) - @test typeof(A_grassmann_hor) <: GrassmannLieAlgHorMatrix{T} - @test eltype(A_grassmann_hor) == T -end - -test_random_array_generation(5, 10, Float32) \ No newline at end of file diff --git a/test/arrays/scalar_multiplication_for_custom_arrays.jl b/test/arrays/scalar_multiplication_for_custom_arrays.jl deleted file mode 100644 index 6d90cff90..000000000 --- a/test/arrays/scalar_multiplication_for_custom_arrays.jl +++ /dev/null @@ -1,43 +0,0 @@ -using GeometricMachineLearning, Test -import Random - -Random.seed!(1234) - -@doc raw""" -This function tests scalar multiplication for various custom arrays, i.e. if \((A,\alpha) \mapsto \alpha{}A\) is performed in the correct way. -""" -function scalar_multiplication_for_custom_arrays(n::Int, N::Int, T::Type) - A = rand(T, n, n) - α = rand(T) - - # SymmetricMatrix - Aα_sym = SymmetricMatrix(α * A) - Aα_sym2 = α * SymmetricMatrix(A) - @test Aα_sym ≈ Aα_sym2 - @test typeof(Aα_sym) <: SymmetricMatrix{T} - @test typeof(Aα_sym2) <: SymmetricMatrix{T} - - # SkewSymMatrix - Aα_skew = SkewSymMatrix(α * A) - Aα_skew2 = α * SkewSymMatrix(A) - @test Aα_skew ≈ Aα_skew2 - @test typeof(Aα_skew) <: SkewSymMatrix{T} - @test typeof(Aα_skew2) <: SkewSymMatrix{T} - - C = rand(T, N, N) - - # StiefelLieAlgHorMatrix - Cα_slahm = StiefelLieAlgHorMatrix(α * C, n) - Cα_slahm2 = α * StiefelLieAlgHorMatrix(C, n) - @test Cα_slahm ≈ Cα_slahm2 - @test typeof(Cα_slahm) <: StiefelLieAlgHorMatrix{T} - @test typeof(Cα_slahm2) <: StiefelLieAlgHorMatrix{T} - - Cα_glahm = GrassmannLieAlgHorMatrix(α * C, n) - Cα_glahm2 = α * GrassmannLieAlgHorMatrix(C, n) - @test Cα_glahm ≈ Cα_glahm2 - @test typeof(Cα_glahm) <: GrassmannLieAlgHorMatrix{T} - @test typeof(Cα_glahm2) <: GrassmannLieAlgHorMatrix{T} -end - -scalar_multiplication_for_custom_arrays(5, 10, Float32) \ No newline at end of file diff --git a/test/arrays/skew_sym_conv_test.jl b/test/arrays/skew_sym_conv_test.jl deleted file mode 100644 index 292723257..000000000 --- a/test/arrays/skew_sym_conv_test.jl +++ /dev/null @@ -1,23 +0,0 @@ -using Zygote -import Random - -Random.seed!(123) - -function test_skew_symmetric_matrix_convergence(n::Int = 5, T::Type = Float32) - A = rand(T, n, n) - A = .5 * (A - A') - ps = (weight = rand(SkewSymMatrix{T}, n), ) - o = Optimizer(AdamOptimizer(), ps) - _loss(ps::NamedTuple, A::AbstractMatrix) = norm(ps.weight - A) - for _ in 1:1200 - o.step += 1 - dp = Zygote.gradient(ps -> _loss(ps, A), ps)[1] - update!(o, o.cache.weight, dp.weight) - ps.weight .= ps.weight + dp.weight - end - # check if type stays SkewSymMatrix - @test typeof(ps.weight) <: SkewSymMatrix - @test ps.weight ≈ A -end - -test_skew_symmetric_matrix_convergence() \ No newline at end of file diff --git a/test/arrays/symmetric_matrix.jl b/test/arrays/symmetric_matrix.jl deleted file mode 100644 index b53ed98f8..000000000 --- a/test/arrays/symmetric_matrix.jl +++ /dev/null @@ -1,29 +0,0 @@ -using GeometricMachineLearning - -using Test -import ChainRulesTestUtils - -function test_multiplication(n::Int=5, T=Float32) - A = rand(SymmetricMatrix{T}, n) - b = rand(T, n) - B = rand(T, n, n) - # test if the custom multiplication is performed the right way - @test A*b == Matrix{T}(A)*b - @test A*B == Matrix{T}(A)*B -end - -function test_calling_symmetric_matrix(n::Int=5, T=Float32) - B = rand(T, n, n) - @test isapprox(SymmetricMatrix(B), .5*(B + B')) -end - -function test_pullback_routine(n::Int=5, T=Float32) - A = rand(SymmetricMatrix{T}, n) - B = rand(T, n, n) - - @test ChainRulesTestUtils.rrule(*, A, B) -end - -test_multiplication() -# this test is not working - problem has to do with FiniteDifferences.jl (I don't know if it's worth looking into this) -# test_calling_symmetric_matrix() \ No newline at end of file diff --git a/test/arrays/test_grassmann_lie_alg_hor_constructors.jl b/test/arrays/test_grassmann_lie_alg_hor_constructors.jl deleted file mode 100644 index 12f1b9bc4..000000000 --- a/test/arrays/test_grassmann_lie_alg_hor_constructors.jl +++ /dev/null @@ -1,47 +0,0 @@ -using GeometricMachineLearning -using Test -import Random - -Random.seed!(123) - -function test_constructors(N::Integer, n::Integer; T::DataType=Float32) - B = rand(T, N - n, n) - B1 = GrassmannLieAlgHorMatrix(B, N, n) - - B2 = Matrix(B1) # note that this does not have any special structure - - B2 = GrassmannLieAlgHorMatrix(B2, n) - - E = StiefelProjection(B1) - - B3 = B1 * E - - B3 = GrassmannLieAlgHorMatrix(B3, n) - - @test B1 ≈ B2 ≈ B3 -end - -function test_lift(N::Integer, n::Integer; T::DataType=Float32) - Y = rand(GrassmannManifold{T}, N, n) - Δ = rgrad(Y, rand(T, N, n)) - ΩΔ = GeometricMachineLearning.Ω(Y, Δ) - λY = GlobalSection(Y) - - λY_mat = Matrix(λY) - - Δ_lift1 = λY_mat' * ΩΔ * λY_mat - - Δ_lift2 = global_rep(λY, Δ) - - @test Δ_lift1 ≈ Δ_lift2 - @test ΩΔ * Y.A ≈ Δ -end - -for T in (Float32, Float64) - for N in (10, 20) - for n in (3, 5) - test_constructors(N, n; T = T) - test_lift(N, n; T = T) - end - end -end \ No newline at end of file diff --git a/test/arrays/test_stiefel_lie_alg_hor_constructors.jl b/test/arrays/test_stiefel_lie_alg_hor_constructors.jl deleted file mode 100644 index 7fc9d4ed2..000000000 --- a/test/arrays/test_stiefel_lie_alg_hor_constructors.jl +++ /dev/null @@ -1,48 +0,0 @@ -using GeometricMachineLearning -using Test -import Random - -Random.seed!(1234) - -function test_constructors(N::Integer, n::Integer; T::DataType=Float32) - A = rand(SkewSymMatrix{T}, n) - B = rand(T, N - n, n) - B1 = StiefelLieAlgHorMatrix(A, B, N, n) - - B2 = Matrix(B1) # note that this does not have any special structure - - B2 = StiefelLieAlgHorMatrix(B2, n) - - E = StiefelProjection(B1) - - B3 = B1 * E - - B3 = StiefelLieAlgHorMatrix(B3, n) - - @test B1 ≈ B2 ≈ B3 -end - -function test_lift(N::Integer, n::Integer; T::DataType=Float32) - Y = rand(StiefelManifold{T}, N, n) - Δ = rgrad(Y, rand(T, N, n)) - ΩΔ = GeometricMachineLearning.Ω(Y, Δ) - λY = GlobalSection(Y) - - λY_mat = Matrix(λY) - - Δ_lift1 = λY_mat' * ΩΔ * λY_mat - - Δ_lift2 = global_rep(λY, Δ) - - @test Δ_lift1 ≈ Δ_lift2 - @test ΩΔ * Y.A ≈ Δ -end - -for T in (Float32, Float64) - for N in (10, 20) - for n in (3, 5) - test_constructors(N, n; T=T) - test_lift(N, n; T=T) - end - end -end \ No newline at end of file diff --git a/test/arrays/triangular.jl b/test/arrays/triangular.jl index cdd264923..9acdbfc87 100644 --- a/test/arrays/triangular.jl +++ b/test/arrays/triangular.jl @@ -1,25 +1,12 @@ -using GeometricMachineLearning +using GeometricMachineLearning using GeometricMachineLearning: mat_tensor_mul -using LinearAlgebra: tr using Zygote: pullback using Test -function triangular_assignment_test(T=Float64, n::Int=5) - A = rand(T, n, n) - LT = LowerTriangular(A) - UT = UpperTriangular(A) - - @test tr(A) ≈ sum(A - LT - UT) -end - -function triangular_multiplication_test(T=Float64, n::Int=5) - Aₗ = rand(LowerTriangular{T}, n) - Aᵤ = rand(UpperTriangular{T}, n) - - B = rand(T, n, n) - @test Aₗ * B ≈ Matrix{T}(Aₗ) * B - @test Aᵤ * B ≈ Matrix{T}(Aᵤ) * B -end +# What the triangular types *are* — their storage layout, their arithmetic, their multiplication +# against a dense matrix — is tested in GeometricOptimizers, which defines them +# (`test/special_matrices/triangular.jl` there). What is left here is GML's: batching them over the +# third axis of a tensor with `mat_tensor_mul`, and the pullback of that kernel. function triangular_tensor_multiplication_test(T=Float64, n::Int=5) Aₗ = rand(LowerTriangular{T}, n) @@ -36,7 +23,7 @@ end function triangular_tensor_multiplication_pullback_test(T=Float64, n::Int=5) Aₗ = rand(LowerTriangular{T}, n) - Aᵤ = rand(LowerTriangular{T}, n) + Aᵤ = rand(UpperTriangular{T}, n) B = rand(T, n, n, n) C_diff = rand(T, n, n, n) @@ -44,17 +31,13 @@ function triangular_tensor_multiplication_pullback_test(T=Float64, n::Int=5) total_pb_lower = pullback(mat_tensor_mul, Aₗ, B)[2](C_diff) total_pb_upper = pullback(mat_tensor_mul, Aᵤ, B)[2](C_diff) + # The batched pullback has to agree slice by slice with the pullback of the single-slice + # product. These were bare expressions and not `@test`s before, so the loop asserted nothing. for i in axes(total_pb_lower[2], 3) - total_pb_lower[2][:, :, i] ≈ pullback(*, Aₗ, B[:, :, i])[2](C_diff[:, :, i])[2] - total_pb_upper[2][:, :, i] ≈ pullback(*, Aᵤ, B[:, :, i])[2](C_diff[:, :, i])[2] + @test total_pb_lower[2][:, :, i] ≈ pullback(*, Aₗ, B[:, :, i])[2](C_diff[:, :, i])[2] + @test total_pb_upper[2][:, :, i] ≈ pullback(*, Aᵤ, B[:, :, i])[2](C_diff[:, :, i])[2] end end -triangular_assignment_test() -triangular_multiplication_test() triangular_tensor_multiplication_test() triangular_tensor_multiplication_pullback_test() - -M = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16] -@test vec(LowerTriangular(M)) == [5, 9, 10, 13, 14, 15] -@test vec(SkewSymMatrix(M)) ≈ [1.5, 3.0, 1.5, 4.5, 3.0, 1.5] diff --git a/test/data_loader/batch_data_loader_qp_test.jl b/test/data_loader/batch_data_loader_qp_test.jl index f968932cf..13047e99b 100644 --- a/test/data_loader/batch_data_loader_qp_test.jl +++ b/test/data_loader/batch_data_loader_qp_test.jl @@ -29,8 +29,8 @@ function test_data_loader(dim=2, number_of_time_steps=100, number_of_parameters= o₁ = Optimizer(GradientOptimizer(), nn1) o₂ = Optimizer(GradientOptimizer(), nn2) - o₁(nn1, dl1, batch) - o₂(nn2, dl2, batch) + o₁(nn1, dl1, batch; show_progress = false) + o₂(nn2, dl2, batch; show_progress = false) end test_data_loader() \ No newline at end of file diff --git a/test/data_loader/data_loader_for_input_and_output.jl b/test/data_loader/data_loader_for_input_and_output.jl index 0ee3c7f29..02acd65c4 100644 --- a/test/data_loader/data_loader_for_input_and_output.jl +++ b/test/data_loader/data_loader_for_input_and_output.jl @@ -21,5 +21,5 @@ batch = Batch(nbatch, 1, 1) loss = FeedForwardLoss() -loss_array = o(nn, dl, batch, nepochs, loss) +loss_array = o(nn, dl, batch, nepochs, loss; show_progress = false) @test loss_array[end] < 0.9 \ No newline at end of file diff --git a/test/manifolds/grassmann_manifold.jl b/test/manifolds/grassmann_manifold.jl deleted file mode 100644 index e1b234905..000000000 --- a/test/manifolds/grassmann_manifold.jl +++ /dev/null @@ -1,73 +0,0 @@ -""" -Warning: all these tests seem to be fine for double precision, but require a ridicolously high tolerance (~5f-3) for single precision! -""" - -using Test -using LinearAlgebra -using GeometricMachineLearning -using GeometricMachineLearning: Ω -using GeometricMachineLearning: global_section -import Random - -Random.seed!(1234) - -function check_gradient(T, N::Integer, n::Integer) - Y = rand(GrassmannManifold{T}, N, n) - - #element of the tangent space - Δ = rgrad(Y, randn(T, N, n)) - A = randn(T, N, n) - V = rgrad(Y, A) - norm(tr(Δ'*A) - metric(Y, Δ, V))/N/n -end - -function global_section_test(T, N::Integer, n::Integer) - Y = rand(GrassmannManifold{T}, N, n) - Q = Matrix(GlobalSection(Y)) - πQ = Q[1:N, 1:n] - norm(Y - πQ * πQ' * Y) / N / n -end - -function tangent_space_rep(T, N::Integer, n::Integer) - Y = rand(GrassmannManifold{T}, N, n) - Δ = rgrad(Y, randn(T, N, n)) - Y.A' * Δ -end - -function gloabl_tangent_space_representation(T, N::Integer, n::Integer) - Y = rand(GrassmannManifold{T}, N, n) - Δ = rgrad(Y, randn(T, N, n)) - λY = GlobalSection(Y) - global_rep(λY, Δ) -end - -function coordinate_chart_rep(T, N::Integer, n::Integer) - Y = rand(GrassmannManifold{T}, N, n) - Y.A = Y.A*inv(Y.A[1:n, 1:n]) - Y -end - -function metric_test(T, N, n) - Y = rand(GrassmannManifold{T}, N, n) - Δ₁ = rgrad(Y, rand(T, N, n)) - Δ₂ = rgrad(Y, rand(T, N, n)) - @test T(.5) * tr(Ω(Y, Δ₁)' * Ω(Y, Δ₂)) ≈ metric(Y, Δ₁, Δ₂) -end - -function run_tests(T, N, n, tol) - @test check_gradient(T, N, n) < tol - @test global_section_test(T, N, n) < tol - @test norm(tangent_space_rep(T, N, n)[1:n,1:n])/N/n < tol - @test typeof(gloabl_tangent_space_representation(T, N, n)) <: GrassmannLieAlgHorMatrix - # because of the matrix inversion the tolerance here is set to a higher value - @test norm(coordinate_chart_rep(T, N, n)[1:n,1:n]-I(n)) / N / n < tol*10 - metric_test(T, N, n) -end - -tol = 1e-8 -T = Float64 -for N in 1:10 - for n in 1:(N-1) - run_tests(T, N, n, tol) - end -end diff --git a/test/manifolds/stiefel_manifold.jl b/test/manifolds/stiefel_manifold.jl deleted file mode 100644 index 42c918474..000000000 --- a/test/manifolds/stiefel_manifold.jl +++ /dev/null @@ -1,65 +0,0 @@ -using Test -using LinearAlgebra -using GeometricMachineLearning -using GeometricMachineLearning: Ω -import Random - -Random.seed!(123) - -N = 5 -A = rand(N,N) -A_skew = SkewSymMatrix(A) - -for i in 1:N - for j in 1:N - @test abs(.5*(A - A')[i,j] - A_skew[i,j]) < 1e-10 - end -end - -n = 1 -A_hor = StiefelLieAlgHorMatrix(A_skew, n) - -for i in 1:n - for j in 1:N - @test abs(A_hor[i,j] - A_skew[i,j]) < 1e-10 - end -end - -for i in (n+1):N - for j in 1:n - @test abs(A_hor[i,j] - A_skew[i,j]) < 1e-10 - end - for j in (n+1):N - @test abs(A_hor[i,j]) < 1e-10 - end -end - -function Ω_test(N::Integer, n::Integer, T::Type=Float32) - Y = rand(StiefelManifold{Float32}, 5, 3) - Δ = rgrad(Y, rand(Float32, 5, 3)) - @test GeometricMachineLearning.Ω(Y, Δ) * Y.A ≈ Δ -end - -function retraction_test(N::Integer, n::Integer, T::Type=Float32) - Y = rand(StiefelManifold{T}, N, n) - Δ = rgrad(Y, rand(T, N, n)) - Y₁ = geodesic(Y, Δ / 1000) - @test norm(1000 * (Y₁ - Y) - Δ) / norm(Δ) < 1e-2 -end - -function metric_test(N, n, T) - Y = rand(StiefelManifold{T}, N, n) - Δ₁ = rgrad(Y, rand(T, N, n)) - Δ₂ = rgrad(Y, rand(T, N, n)) - @test T(.5) * tr(Ω(Y, Δ₁)' * Ω(Y, Δ₂)) ≈ metric(Y, Δ₁, Δ₂) -end - -for N in (20, 10) - for n in (5, 3) - for T in (Float64, Float32) - Ω_test(N, n, T) - retraction_test(N, n, T) - metric_test(N, n, T) - end - end -end \ No newline at end of file diff --git a/test/manifolds/symplectic_stiefel_manifold.jl b/test/manifolds/symplectic_stiefel_manifold.jl deleted file mode 100644 index 66573c5bc..000000000 --- a/test/manifolds/symplectic_stiefel_manifold.jl +++ /dev/null @@ -1,30 +0,0 @@ -using GeometricMachineLearning -using GeometricMachineLearning: global_section, Ω -using Quadmath: Float128 - -import LinearAlgebra - -N = 10 -n = 5 - -function symplectic_stiefel_manifold_tests(T, N, n) - U = rand(SymplecticStiefelManifold{T}, 2*N, 2*n) - check_val = check(U) - print("ErrSympl",T,": ", check_val, "\n") - #this is the version using symplectic Householder reflections - S = global_section₂(U) - global_section_error = LinearAlgebra.norm((inv(S)*U)[vcat(1:(N-n), (N+1):(2*N-n)), :]) - print("ErrGlobalSection",T,": ", global_section_error, "\n") - - J = PoissonTensor(eltype(U), N) - Δ = rgrad(U, rand(eltype(U), 2*N, 2*n), J) - print("error in vector space property", T, ": ", LinearAlgebra.norm(Δ'*J*U + U'*J*Δ), "\n") - print("error lie algebra lift", T ,": ", LinearAlgebra.norm(Ω(U, Δ)*U - Δ), "\n") -end - -@time symplectic_stiefel_manifold_tests(Float32, N, n) -print("\n") -@time symplectic_stiefel_manifold_tests(Float64, N, n) -print("\n") -@time symplectic_stiefel_manifold_tests(Float128, N, n) - diff --git a/test/network_losses/losses_and_optimization.jl b/test/network_losses/losses_and_optimization.jl index 9ace7aca4..ecc796dd2 100644 --- a/test/network_losses/losses_and_optimization.jl +++ b/test/network_losses/losses_and_optimization.jl @@ -19,7 +19,7 @@ function train_network(; n_epochs=10) o = Optimizer(Adam(), nn) batch = Batch(5, 1) - loss_array = o(nn, dl, batch, n_epochs, loss) + loss_array = o(nn, dl, batch, n_epochs, loss; show_progress = false) T = eltype(dl) @test loss_array[end] / loss_array[1] < T(0.1) end diff --git a/test/optimizers/exponential_retractions.jl b/test/optimizers/exponential_retractions.jl deleted file mode 100644 index 966cbc31d..000000000 --- a/test/optimizers/exponential_retractions.jl +++ /dev/null @@ -1,41 +0,0 @@ -using Test -using LinearAlgebra -using Printf - -using Random - -using GeometricMachineLearning - -#NOTE: zeros have to be added because exp() is not defined for SkewSymMatrix or StiefelLieAlgHorMatrix!!! -function exponential_retraction₁(Y::StiefelManifold, Δ::AbstractMatrix, η) - StiefelManifold(exp(η*Ω(Y,Δ) - zeros(size(Y,1),size(Y,1)))*Y) -end - -function exponential_retraction₂(Y::StiefelManifold, Δ::AbstractMatrix, η) - N, n = size(Y) - HD, B = global_rep(Y, Δ) - E = StiefelProjection(N, n) - Y₂ = StiefelManifold(exp(η*B - zeros(size(Y,1),size(Y,1)))*E) - apply_λ(Y, HD, Y₂) -end - - -N_vec = 2 .^ collect(7:12) -n_vec = 2 .^ collect(1:6) -ε = 1e-11 -η = .1 - -for (N, n) ∈ zip(N_vec, n_vec) - print("N = "*string(N)*", n = "*string(n)*"\n") - Y = StiefelManifold(N, n) - Δ = SkewSymMatrix(N)*Y - @printf "Standard exponential: " - @time sol₁ = exponential_retraction₁(Y, Δ, η) - @printf "Exponential with householder (expected to be slower):" - @time sol₂ = exponential_retraction₂(Y, Δ, η) - @printf "Custom implementation (also gives householder): " - @time sol₃ = Exp(Y, Δ, η) - @test norm(sol₁ - sol₂) < ε - @test norm(sol₁ - sol₃) < ε - print("\n") -end \ No newline at end of file diff --git a/test/optimizers/hor_lift.jl b/test/optimizers/hor_lift.jl deleted file mode 100644 index 3089f3326..000000000 --- a/test/optimizers/hor_lift.jl +++ /dev/null @@ -1,4 +0,0 @@ -include("../src/arrays/sympl_st_E_ts.jl") -include("../src/arrays/sympl_lie_alg_hor.jl") -include("../src/arrays/symplectic_lie_alg2.jl") -include("../src/optimizers/auxiliary_gradients.jl") diff --git a/test/optimizers/lie_alg_lifts.jl b/test/optimizers/lie_alg_lifts.jl deleted file mode 100644 index bc1da8f7f..000000000 --- a/test/optimizers/lie_alg_lifts.jl +++ /dev/null @@ -1,21 +0,0 @@ -""" -This function tests if the lift really maps to the invariant horizontal component of the Lie algebra. -""" - -using LinearAlgebra -using Test - -include("../src/arrays/skew_sym.jl") -include("../src/optimizers/householder.jl") -include("../src/optimizers/manifold_types.jl") -include("../src/arrays/stiefel_lie_alg_hor.jl") -include("../src/optimizers/lie_alg_lifts.jl") -include("../src/arrays/auxiliary.jl") - -function stiefel_lift_test(N, n, ε=1e-12) - Y = StiefelManifold(N, n) - V = SkewSymMatrix(randn(N,N))*Y - #global_rep gives two outputs: a householder elment and the lifted element of the Lie algebra - V_lift = global_rep_test(Y,V)[2] - @test norm(V_lift - StiefelLieAlgHorMatrix(SkewSymMatrix(V_lift), n))/N < ε -end \ No newline at end of file diff --git a/test/optimizers/manifold_optim.jl b/test/optimizers/manifold_optim.jl deleted file mode 100644 index cb1f7d7c7..000000000 --- a/test/optimizers/manifold_optim.jl +++ /dev/null @@ -1,41 +0,0 @@ -using GeometricMachineLearning -using Lux -using Random -using Zygote -using LinearAlgebra - -m = 20 -n = 200 -x = rand(2 * n) - -model = Chain(Gradient(2 * n, 4 * n), Gradient(2 * n), SymplecticStiefelLayer(2 * m, 2 * n; inverse = true)) - -n_runs = Int(5e3) -err_vec = zeros(n_runs+1) -err_vec2 = zeros(n_runs+1) - -### Test for StandardOptimizer -#note! optimizer and network state are not the same! -optim = StandardOptimizer(1e-5) -ps, st = Lux.setup(Random.default_rng(), model) -err_vec[1] = norm(Lux.apply(model,x,ps,st)[1]) -@time for i in 1:n_runs - g = gradient(p -> norm(Lux.apply(model, x, p, st)[1]), ps)[1] - apply!(optim, nothing, model, ps, g) - err_vec[i+1] = norm(Lux.apply(model,x,ps,st)[1]) -end -print(norm(Lux.apply(model,x,ps,st)[1])) - -### Test for MomentumOptimizer -optim = MomentumOptimizer(1e-5,1e-2) -ps, st = Lux.setup(Random.default_rng(), model) -#hacky for the moment!!!!!!!!!1 fix!!!!! -model2 = Chain(Gradient(2 * n, 4 * n), Gradient(2 * n), SymplecticStiefelLayer(2 * n, 2 * n; inverse = true)) -state = init_momentum(model2) -err_vec2[1] = norm(Lux.apply(model,x,ps,st)[1]) -@time for i in 1:n_runs - g = gradient(p -> norm(Lux.apply(model, x, p, st)[1]), ps)[1] - apply!(optim, state, model, ps, g) - err_vec2[i+1] = norm(Lux.apply(model,x,ps,st)[1]) -end -print(norm(Lux.apply(model,x,ps,st)[1])) diff --git a/test/optimizers/manifold_related/legacy_functions.jl b/test/optimizers/manifold_related/legacy_functions.jl deleted file mode 100644 index 501a4099c..000000000 --- a/test/optimizers/manifold_related/legacy_functions.jl +++ /dev/null @@ -1,27 +0,0 @@ -""" -These may be useful for testing purposes, but are no longer used in src. -""" - - -#I might actually not need this! -function Ω(U::SymplecticStiefelManifold{T}, Δ::AbstractMatrix{T}) where {T} - J_mat = PoissonTensor(T, size(U,1)÷2) - SymplecticLieAlgMatrix( - Δ*inv(U'*U)*U' + J_mat*U*inv(U'U)*Δ'*(I + J_mat*U*inv(U'*U)*U'*J_mat)*J_mat - ) -end - -Ω₁(Y::StiefelManifold, Δ::AbstractMatrix) = SkewSymMatrix(2*(I - .5*Y*Y')*Δ*Y') -#TODO: perform calculations in-place, don't allocate so much! -function Ω(Y::StiefelManifold, Δ::AbstractMatrix) - N = size(Y,1) - B̃ = zeros(N, N) - mul!(B̃, Δ, Y') - B̂ = zero(B̃) - mul!(B̂, Y, Y') - rmul!(B̂, -.5) - @views B̂ .+= one(B̂) - B = zero(B̂) - mul!(B, B̂, B̃) - SkewSymMatrix(B) -end \ No newline at end of file diff --git a/test/optimizers/momentum_optim_test.jl b/test/optimizers/momentum_optim_test.jl deleted file mode 100644 index 9e1f11bf3..000000000 --- a/test/optimizers/momentum_optim_test.jl +++ /dev/null @@ -1,29 +0,0 @@ -using GeometricMachineLearning -using Lux -using Random -using Zygote -using Printf - -model = Lux.Chain(Lux.Dense(4, 3), Lux.Dense(3, 1)) -ps, st = Lux.setup(Random.default_rng(), model) - -random_element = randn(4) -function f(p) - sum(model(random_element, p, st)[1])^2 -end - -old_val = f(ps) - -optim = MomentumOptimizer(1e-3,5e-1) -g = Zygote.gradient(f, ps)[1] -cache = MomentumOptimizerCache(optim, model, ps, g) - -post_init_val = f(ps) - -apply!(optim, cache, model, ps, g) -new_val = f(ps) -@printf "Before optimization: %.5e. " old_val -@printf "After initialization: %.5e. " post_init_val -@printf "After optimization: %.5e. " new_val - - diff --git a/test/optimizers/optimizer_convergence_tests/adam_with_learning_rate_decay.jl b/test/optimizers/optimizer_convergence_tests/adam_with_learning_rate_decay.jl index b852c4585..600a4dd73 100644 --- a/test/optimizers/optimizer_convergence_tests/adam_with_learning_rate_decay.jl +++ b/test/optimizers/optimizer_convergence_tests/adam_with_learning_rate_decay.jl @@ -21,13 +21,17 @@ function train_network(; n_epochs=2048) nn₂ = setup_network(dl) o₁ = Optimizer(AdamOptimizer(), nn₁) - o₂ = Optimizer(AdamOptimizerWithDecay(n_epochs), nn₂) + # `AdamOptimizerWithDecay` is GeometricOptimizers' now, and returns the `(algorithm, linesearch)` + # pairing its own `Optimizer` takes, so it splats rather than being passed positionally. The + # element type is positional and defaults to `Float64` here, where GML's own version took it + # from `η₁` and so defaulted to `Float32`. + o₂ = Optimizer(nn₂; AdamOptimizerWithDecay(n_epochs, eltype(dl))...) batch = Batch(5, 1) loss = FeedForwardLoss() - loss_array₁ = o₁(nn₁, dl, batch, n_epochs, loss) - loss_array₂ = o₂(nn₂, dl, batch, n_epochs, loss) + loss_array₁ = o₁(nn₁, dl, batch, n_epochs, loss; show_progress = false) + loss_array₂ = o₂(nn₂, dl, batch, n_epochs, loss; show_progress = false) T = eltype(dl) @test loss_array₂[end] < loss_array₁[end] < T(1.6e-1) @@ -36,15 +40,18 @@ end @doc raw""" `AdamOptimizerWithDecay` also has to work with manifold weights. -It is not one of `GeometricOptimizers`' own methods, so it needs to be routed onto GO's Adam cache -explicitly; without that every weight -- including the `StiefelManifold` ones -- falls through to the -Euclidean state, whose zero element is a `StiefelLieAlgHorMatrix` and not a manifold point. +The pairing is `Adam` plus a `DecayingStatic` step size, so the method the optimizer dispatches on is +an ordinary `Adam` and the manifold weights take GeometricOptimizers' Adam cache like any other Adam. +GML's own version of this method was a distinct `OptimizerMethod` that had to be routed onto that +cache explicitly, and without the routing every weight -- the `StiefelManifold` ones included -- fell +through to the Euclidean state, whose zero element is a `StiefelLieAlgHorMatrix` and not a manifold +point. """ function train_manifold_network(; n_epochs = 32) arch = Chain(StiefelLayer(1, 20), Dense(20, 20, tanh), Dense(20, 1, identity)) nn = NeuralNetwork(arch, CPU(), eltype(dl)) - o = Optimizer(AdamOptimizerWithDecay(n_epochs), nn) + o = Optimizer(nn; AdamOptimizerWithDecay(n_epochs, eltype(dl))...) loss_array = o(nn, dl, Batch(5, 1), n_epochs, FeedForwardLoss(); show_progress = false) @test all(isfinite, loss_array) @@ -54,5 +61,38 @@ function train_manifold_network(; n_epochs = 32) @test Y' * Y ≈ I end +@doc raw""" +The schedule is walked from ``t = 1``, not from ``t = 0``. + +`optimization_step!` increments `opt.iterations` *before* it reads the step size, so the first step +of a run takes ``\alpha(1) = \gamma\eta_1`` and not ``\alpha(0) = \eta_1``. That is how the pre-0.5 +`AdamOptimizerWithDecay` counted — it incremented `o.step` before `update!` — and how +`DecayingStatic` counts, because `GeometricOptimizers.solve!` calls `increase_iteration_number!` +before `solver_step!`. Reading before incrementing put every step of a run one place early in the +schedule, which `test/adam_optimizer_with_decay.jl` upstream asserts does not happen. +""" +function schedule_starts_at_one(; n_epochs = 100, η₁ = 1e-2, η₂ = 1e-6) + method = AdamOptimizerWithDecay(n_epochs, Float64; η₁ = η₁, η₂ = η₂) + o = Optimizer((weight = zeros(2, 2),); method...) + + γ = exp(log(η₂ / η₁) / n_epochs) + @test o.step_size isa DecayingStatic + @test o.iterations == 0 + + for t in 1:4 + o.iterations += 1 + @test GeometricMachineLearning._current_step_size(o, o.iterations) ≈ η₁ * γ^t + end + + # and the same thing through the public entry point: one step of a Euclidean parameter with a + # gradient of `1` moves it by `α₁ / (√1 + δ) ≈ α₁`, so the distance travelled reports the α used + ps = (weight = zeros(2, 2),) + opt = Optimizer(ps; method...) + optimization_step!(opt, GlobalSection(ps), ps, (weight = ones(2, 2),)) + @test opt.iterations == 1 + @test abs(ps.weight[1, 1]) ≈ η₁ * γ rtol = 1e-6 +end + train_network() train_manifold_network() +schedule_starts_at_one() diff --git a/test/optimizers/riemannian_gradients.jl b/test/optimizers/riemannian_gradients.jl deleted file mode 100644 index 5c0b99144..000000000 --- a/test/optimizers/riemannian_gradients.jl +++ /dev/null @@ -1,36 +0,0 @@ -""" -This implements tests for the Riemannian gradients. - -TODO: find correct expression for Riemannian gradient for the canonical metric! -""" - -using Test -using LinearAlgebra -using GeometricMachineLearning - -#Riemannian metric for the Stiefel manifold -> this is probably not needed explicitly! -function riemannian_metric(Y::StiefelManifold, Δ₁, Δ₂) - tr(Δ₁'*(I - .5*Y*Y')*Δ₂) -end - - -function stiefel_riemannian_gradient_test(N::Int, n::Int, ε = 1e-12) - #sample element from 𝔐 - Y = StiefelManifold(N,n) - #sample element from T𝔐 (tangent space): - V = SkewSymMatrix(randn(N,N))*Y - #sample element from T*𝔐 (cotangent space): - A = randn(N,n) - @test norm(tr(A'*V) - metric(Y, rgrad(Y, A), V)) < ε -end - -N_max = 20 -n_max = 10 -num = 10 -N_vec = Int.(ceil.(rand(num)*N_max)) -n_vec = Int.(ceil.(rand(num)*n_max)) -n_vec = min.(n_vec, N_vec) - -for (N, n) ∈ zip(N_vec, n_vec) - stiefel_riemannian_gradient_test(N, n) -end \ No newline at end of file diff --git a/test/optimizers/standard_optim_test.jl b/test/optimizers/standard_optim_test.jl deleted file mode 100644 index 2af65aaf4..000000000 --- a/test/optimizers/standard_optim_test.jl +++ /dev/null @@ -1,34 +0,0 @@ -using GeometricMachineLearning -using Lux -using Random -using Zygote -using Printf - -model = Lux.Chain(Lux.Dense(4, 3), Lux.Dense(3, 1)) -ps, st = Lux.setup(Random.default_rng(), model) - -random_element = randn(4) -function f(p) - sum(model(random_element, p, st)[1])^2 -end - -old_val = f(ps) -optim = StandardOptimizer(1e-3) -g = Zygote.gradient(f, ps)[1] -ps1 = deepcopy(ps) - -#This has to be changed to work with the new optimizers syntax! -#= -apply!(optim, nothing, model, ps1, g) -new_val = f(ps1) -@printf "Before optimization: %.5e. " old_val -@printf "After optimization: %.5e" new_val - -for layer_number in 1:length(model) - for key in keys(ps[layer_number]) - ps[layer_number][key] .-= 1e-3 * g[layer_number][key] - end -end -new_val_manual = f(ps) -@printf "After manuel optimization %.5e" new_val_manual -=# \ No newline at end of file diff --git a/test/optimizers/structured_array_parameters.jl b/test/optimizers/structured_array_parameters.jl index b5e0e00ce..81ee86cfc 100644 --- a/test/optimizers/structured_array_parameters.jl +++ b/test/optimizers/structured_array_parameters.jl @@ -10,20 +10,31 @@ Every optimizer method has to work with the structured matrix types GML uses as `SymmetricMatrix` (SympNet and symplectic attention layers), `SkewSymMatrix` (volume-preserving attention) and `LowerTriangular`/`UpperTriangular` (volume-preserving feedforward layers). -These have their own storage and no `setindex!`, so they need type-preserving `similar` and the -elementwise bridges in `src/optimizers/go_bridges.jl`; without them the optimizer cache cannot even -be allocated. +These have their own storage and no `setindex!`, so they need a type-preserving `similar` and +elementwise `_add!`/`_rac!`/`_square!`/`_div!`/`_rmul!` written on the free parameters; without them +the optimizer cache cannot even be allocated. Those live in GeometricOptimizers, which owns the +types — see `VectorStorageMatrix` there. GML used to carry its own copies of both the types and the +methods, in `src/optimizers/go_bridges.jl`. """ function optimizer_runs(architecture, batch, input_dim; T = Float64, n_epochs = 2) - for method in (AdamOptimizer(), MomentumOptimizer(), GradientOptimizer(), - AdamOptimizerWithDecay(n_epochs)) - nn = NeuralNetwork(architecture, T) - dl = DataLoader(rand(T, input_dim, 20, 5); suppress_info = true) + nn_and_dl() = (NeuralNetwork(architecture, T), + DataLoader(rand(T, input_dim, 20, 5); suppress_info = true)) + + for method in (AdamOptimizer(), MomentumOptimizer(), GradientOptimizer()) + nn, dl = nn_and_dl() o = Optimizer(method, nn) loss_array = o(nn, dl, batch, n_epochs; show_progress = false) @test length(loss_array) == n_epochs @test all(isfinite, loss_array) end + + # `AdamOptimizerWithDecay` is a `(algorithm, linesearch)` pairing rather than a method, so it + # goes in through the keyword constructor + nn, dl = nn_and_dl() + o = Optimizer(nn; AdamOptimizerWithDecay(n_epochs, T)...) + loss_array = o(nn, dl, batch, n_epochs; show_progress = false) + @test length(loss_array) == n_epochs + @test all(isfinite, loss_array) end # the cache has to keep the parameter's type, or the optimizer allocates dense scratch arrays and diff --git a/test/optimizers/utils/global_sections.jl b/test/optimizers/utils/global_sections.jl deleted file mode 100644 index c39729645..000000000 --- a/test/optimizers/utils/global_sections.jl +++ /dev/null @@ -1,37 +0,0 @@ -using GeometricMachineLearning -using LinearAlgebra -using Test - -function global_stiefel_section(N, n) - Y = rand(StiefelManifold, N, n) - λY = GlobalSection(Y) - - E = zeros(N, n) - for i = 1:n - E[i, i] = 1. - end - E = StiefelManifold(E) - Y2 = apply_section(λY, E) - @test typeof(Y2) <: StiefelManifold - @test isapprox(Y2, Y) -end - -function global_tangent_space_rep(N, n) - Y = rand(StiefelManifold, N, n) - λY = GlobalSection(Y) - - Δ = rgrad(Y, rand(N, n)) - B = global_rep(λY, Δ) - BE = B*StiefelProjection(N, n) - # abuse of notation - Δ₂ = typeof(Δ)(apply_section(λY, StiefelManifold(BE))) - @test isapprox(Δ₂, Δ) -end - -N_max = 10 -for N = 2:N_max - for n = 2:N - global_stiefel_section(N, n) - global_tangent_space_rep(N, n) - end -end \ No newline at end of file diff --git a/test/runtests.jl b/test/runtests.jl index 102cd89d2..d8758cd8d 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -1,5 +1,10 @@ using SafeTestsets, Test, GeometricMachineLearning +# A test that trains passes `show_progress = false`. The `Optimizer` functor defaults it to `true`, +# which is right at a REPL and is noise in a suite -- a 2048-epoch run emits a few hundred progress +# lines and buries the failure you are looking for. `train!` already defaults `showprogress = false`, +# so only the functor needs saying. + # reduced order modeling tests @info "Starting reduced-order-modeling tests" @safetestset "PSD tests " begin @@ -17,45 +22,12 @@ end @safetestset "Check parameterlength " begin include("parameterlength/check_parameterlengths.jl") end -@safetestset "Arrays #1 " begin - include("arrays/array_tests.jl") -end -@safetestset "Map to skew " begin - include("arrays/map_to_skew.jl") -end -@safetestset "Sampling of arrays " begin - include("arrays/random_generation_of_custom_arrays.jl") -end -@safetestset "Addition tests for custom arrays " begin - include("arrays/addition_tests_for_custom_arrays.jl") -end -@safetestset "Scalar multiplication tests for custom arrays " begin - include("arrays/scalar_multiplication_for_custom_arrays.jl") -end -@safetestset "Matrix multiplication tests for custom arrays " begin - include("arrays/matrix_multiplication_for_custom_arrays.jl") -end -@safetestset "Test constructors for custom arrays " begin - include("arrays/constructor_tests_for_custom_arrays.jl") -end @safetestset "Symplectic Potential (array tests) " begin include("arrays/poisson_tensor.jl") end -@safetestset "Test StiefelLieAlgHorMatrix constructors and lifts " begin - include("arrays/test_stiefel_lie_alg_hor_constructors.jl") -end -@safetestset "Test GrassmannLieAlgHorMatrix constructors and lifts " begin - include("arrays/test_grassmann_lie_alg_hor_constructors.jl") -end @safetestset "Test triangular matrices " begin include("arrays/triangular.jl") end -@safetestset "Manifolds (Stiefel): " begin - include("manifolds/stiefel_manifold.jl") -end -@safetestset "Manifolds (Grassmann): " begin - include("manifolds/grassmann_manifold.jl") -end @safetestset "Gradient Layer " begin include("layers/gradient_layer_tests.jl") end @@ -108,9 +80,6 @@ end include("layers/classification.jl") end @info "Starting optimizer tests" -@safetestset "Optimizer #1 " begin - include("optimizers/utils/global_sections.jl") -end @safetestset "Optimizer #2 " begin include("optimizers/utils/optimization_step.jl") end diff --git a/test/sae_error_lower_than_psd_error.jl b/test/sae_error_lower_than_psd_error.jl index 952c18a80..64ea81686 100644 --- a/test/sae_error_lower_than_psd_error.jl +++ b/test/sae_error_lower_than_psd_error.jl @@ -13,7 +13,7 @@ function test_accuracy(N::Integer, n::Integer; tol::Real = .35, n_epochs::Intege sae_nn = NeuralNetwork(SymplecticAutoencoder(N, n; n_encoder_layers = 5, n_decoder_layers = 5)) o = Optimizer(Adam(), sae_nn) - sae_error = o(sae_nn, dl, Batch(10), n_epochs)[end] + sae_error = o(sae_nn, dl, Batch(10), n_epochs; show_progress = false)[end] @test sae_error < psd_error end diff --git a/test/symplectic_autoencoder_tests.jl b/test/symplectic_autoencoder_tests.jl index 2006ef0f9..171edf8a4 100644 --- a/test/symplectic_autoencoder_tests.jl +++ b/test/symplectic_autoencoder_tests.jl @@ -11,7 +11,7 @@ function test_accuracy(N::Integer, n::Integer; tol::Real = 0.35, n_epochs::Integ sae_nn = NeuralNetwork(SymplecticAutoencoder(N, n)) o = Optimizer(Adam(), sae_nn) - sae_error = o(sae_nn, dl, Batch(10), n_epochs)[end] + sae_error = o(sae_nn, dl, Batch(10), n_epochs; show_progress = false)[end] @test sae_error < tol end @@ -40,7 +40,7 @@ function test_symplecticity(N::Integer, n::Integer) # test if it's still symplectic after training dl = DataLoader(rand(N, 10 * N); autoencoder = true) o = Optimizer(Adam(), sae_nn) - o(sae_nn, dl, Batch(10), 10) + o(sae_nn, dl, Batch(10), 10; show_progress = false) sympl_mat = jacobian(vec -> sae_decoder(vec), test_vector)[1] @test PoissonTensor(n) ≈ sympl_mat' * PoissonTensor(N) * sympl_mat end diff --git a/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl b/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl index 00da1ef94..0b2bc8224 100644 --- a/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl +++ b/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl @@ -2,6 +2,7 @@ using GeometricMachineLearning, Test # qualified access only: a blanket `using` would make `StiefelManifold` and friends ambiguous, # since GeometricMachineLearning re-exports its own versions of them import GeometricOptimizers +using GeometricOptimizers: GradientCache, MomentumCache, AdamCache import Random, LinearAlgebra Random.seed!(1234) diff --git a/test/transformer_related/multi_head_attention_stiefel_retraction.jl b/test/transformer_related/multi_head_attention_stiefel_retraction.jl index 76fabf4c1..b007dd66b 100644 --- a/test/transformer_related/multi_head_attention_stiefel_retraction.jl +++ b/test/transformer_related/multi_head_attention_stiefel_retraction.jl @@ -1,6 +1,9 @@ using GeometricMachineLearning, Test using GeometricMachineLearning: geodesic using GeometricMachineLearning: cayley +# The optimizer caches are internal to GeometricOptimizers -- they are `solver_step!` scratch -- +# so they are named qualified rather than through a re-export. +using GeometricOptimizers: MomentumCache import Random, Test, LinearAlgebra, KernelAbstractions Random.seed!(1234) diff --git a/test/transformer_related/multi_head_attention_stiefel_setup.jl b/test/transformer_related/multi_head_attention_stiefel_setup.jl index 3798c08c8..6573c146a 100644 --- a/test/transformer_related/multi_head_attention_stiefel_setup.jl +++ b/test/transformer_related/multi_head_attention_stiefel_setup.jl @@ -1,4 +1,6 @@ using GeometricMachineLearning, Test +# see the note in `multi_head_attention_stiefel_retraction.jl` +using GeometricOptimizers: MomentumCache import Random, Test, LinearAlgebra, KernelAbstractions Random.seed!(1234) From 85c515272614112e6b2ab8820271b5a10e4c5411 Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Mon, 17 Aug 2026 21:58:52 +0900 Subject: [PATCH 03/12] Hand the manifold and optimizer chapters to GeometricOptimizers MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Thirteen pages leave. They documented types that live in GeometricOptimizers and, after the first commit of this branch, are not even defined here any more. Manifolds (all 7 pages) general topology through homogeneous spaces arrays/skew_symmetric_matrix.md the structured matrix types arrays/global_tangent_spaces.md 𝔤^hor, global sections optimizers/ (all 4 pages) the framework, retractions, parallel transport, the methods They land upstream as a `Manifolds` section, `special_matrices.md`, `global_tangent_spaces.md`, `parallel_transport.md` and `optimizer_methods.md`; `optimizer_framework.md`'s theory merges into `manifold_optimizers.md` and the retraction theory into `retractions.md`. See GeometricOptimizers#50. ## Three pages split rather than moved - **`optimizer_framework.md`** documented GML's own `Optimizer`, `optimize_for_one_epoch!` and `optimization_step!`. The theory goes; a new `docs/src/optimizers/optimizer.md` keeps those three and says what this package adds to the framework -- walking the parameter tree of a `NeuralNetwork` and driving it from a data loader. - **`arrays/skew_symmetric_matrix.md`**'s *Parallel Computation* section documented `tensor_mat_mul`/`mat_tensor_mul`, which are GML's. Folded into `arrays/tensors.md`. - **`arrays/tensors.md`** and **`pullbacks/computation_of_pullbacks.md`** stay whole: GML's kernels and GML's AD. ## Cross-references Thirty-six references from pages that stayed into pages that left, and they stay references: `DocumenterInterLinks` joins `docs/Project.toml` and `make.jl`. This also closes **C3**, which asked for exactly this -- `𝔄`, `cayley` and `update!` had been downgraded to plain code spans as a stopgap. The inventory is read from a **committed file**, `docs/inventories/GeometricOptimizers.toml`, and not from a URL. The anchors this needs only appear in upstream's published `objects.inv` once 0.4.0's documentation deploys, and a documentation build that cannot run until an unrelated deploy has happened is a build that will be broken again later for the same reason. The comment above `links = InterLinks(...)` has the one-line command that regenerates it. Two things about `@extref` targets that cost a build each, recorded so the next person does not pay them again: the target is the inventory's **slug** (`The-Grassmann-Manifold`, not `"The Grassmann Manifold"`), and for a binding it is **module-qualified** (`GeometricOptimizers.SymmetricMatrix`). A bare `[Title](@ref)` with no explicit target is a third form and needs converting too. ## The book loses a part and a chapter `_html_pages` drops `Manifolds` and `Optimizer`; `Special Arrays and AD` keeps two of its four pages. In `_latex_pages` the whole `Manifolds` chapter and the whole `Optimizer` part go, so `docstring_index.md` is rewritten to match and `abstract.md`, `introduction.md` and `outlook.md` no longer promise chapters that are not there -- they point at upstream instead. `_optimizers` is now a single page and goes into `_latex_pages` as a `Pair{String, String}` rather than as a one-element vector: `Dict(_latex_pages)` infers its value type from these entries, and one vector-of-strings among the vector-of-pairs chapters makes that inference fail outright. Co-Authored-By: Claude Opus 5 (1M context) --- docs/Project.toml | 7 +- docs/inventories/GeometricOptimizers.toml | 955 ++++++++++++++++++ docs/make.jl | 70 +- .../linear_symplectic_transformer.md | 2 +- .../neural_network_integrators.md | 2 +- .../architectures/symplectic_autoencoder.md | 2 +- docs/src/architectures/sympnet.md | 4 +- docs/src/arrays/global_tangent_spaces.md | 288 ------ docs/src/arrays/skew_symmetric_matrix.md | 166 --- docs/src/arrays/tensors.md | 2 +- docs/src/docstring_index.md | 20 +- docs/src/index.md | 2 +- docs/src/introduction.md | 22 +- docs/src/layers/attention_layer.md | 6 +- docs/src/layers/symplectic_attention.md | 6 +- docs/src/layers/sympnet_gradient.md | 2 +- .../layers/volume_preserving_feedforward.md | 4 +- docs/src/manifolds/basic_topology.md | 159 --- .../existence_and_uniqueness_theorem.md | 81 -- docs/src/manifolds/homogeneous_spaces.md | 244 ----- .../src/manifolds/inverse_function_theorem.md | 99 -- docs/src/manifolds/manifolds.md | 265 ----- .../src/manifolds/metric_and_vector_spaces.md | 115 --- docs/src/manifolds/riemannian_manifolds.md | 230 ----- .../manifold_related/parallel_transport.md | 232 ----- .../manifold_related/retractions.md | 407 -------- docs/src/optimizers/optimizer.md | 29 + docs/src/optimizers/optimizer_framework.md | 116 --- docs/src/optimizers/optimizer_methods.md | 181 ---- docs/src/outlook.md | 6 +- .../src/pullbacks/computation_of_pullbacks.md | 4 +- docs/src/reduced_order_modeling/losses.md | 2 +- .../pod_autoencoders.md | 2 +- .../reduced_order_modeling/symplectic_mor.md | 4 +- .../structure_preserving_neural_networks.md | 2 +- .../structure_preservation/symplecticity.md | 4 +- docs/src/tutorials/grassmann_layer.md | 2 +- .../linear_symplectic_transformer.md | 8 +- docs/src/tutorials/symplectic_autoencoder.md | 12 +- docs/src/tutorials/symplectic_transformer.md | 8 +- docs/src/tutorials/sympnet_tutorial.md | 2 +- ...olume_preserving_transformer_rigid_body.md | 10 +- 42 files changed, 1092 insertions(+), 2692 deletions(-) create mode 100644 docs/inventories/GeometricOptimizers.toml delete mode 100644 docs/src/arrays/global_tangent_spaces.md delete mode 100644 docs/src/arrays/skew_symmetric_matrix.md delete mode 100644 docs/src/manifolds/basic_topology.md delete mode 100644 docs/src/manifolds/existence_and_uniqueness_theorem.md delete mode 100644 docs/src/manifolds/homogeneous_spaces.md delete mode 100644 docs/src/manifolds/inverse_function_theorem.md delete mode 100644 docs/src/manifolds/manifolds.md delete mode 100644 docs/src/manifolds/metric_and_vector_spaces.md delete mode 100644 docs/src/manifolds/riemannian_manifolds.md delete mode 100644 docs/src/optimizers/manifold_related/parallel_transport.md delete mode 100644 docs/src/optimizers/manifold_related/retractions.md create mode 100644 docs/src/optimizers/optimizer.md delete mode 100644 docs/src/optimizers/optimizer_framework.md delete mode 100644 docs/src/optimizers/optimizer_methods.md diff --git a/docs/Project.toml b/docs/Project.toml index 5f0965c41..b12c8ba9f 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -3,8 +3,10 @@ AbstractNeuralNetworks = "60874f82-5ada-4c70-bd1c-fa6be7711c8a" Bibliography = "f1be7e48-bf82-45af-a471-ae754a193061" BrenierTwoFluid = "698bc5df-bacc-4e45-9592-41ae9e406d75" CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0" +DocInventories = "43dc2714-ed3b-44b5-b226-857eda1aa7de" Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" DocumenterCitations = "daee34ce-89f3-4625-b898-19384cb65244" +DocumenterInterLinks = "d12716ef-a0f6-4df4-a9f1-a5a34e75c656" ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" GeometricIntegrators = "dcce2d33-59f6-5b8d-9047-0defad88ae06" GeometricMachineLearning = "194d25b2-d3f5-49f0-af24-c124f4aa80cc" @@ -14,8 +16,7 @@ HDF5 = "f67ccb44-e63f-5c2f-98bd-6dc0ccc4ba2f" LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" -# GeometricMachineLearning itself is deliberately *not* listed here: `docs/Makefile`'s `test_docs` -# target develops it from the repository root, which is what makes the same file work in CI and in a -# local clone at any path. [sources] BrenierTwoFluid = {rev = "main", url = "https://github.com/ToBlick/BrenierTwoFluids.git"} +GeometricMachineLearning = {path = ".."} +GeometricOptimizers = {path = "../../GeometricOptimizers"} diff --git a/docs/inventories/GeometricOptimizers.toml b/docs/inventories/GeometricOptimizers.toml new file mode 100644 index 000000000..ba64e36a4 --- /dev/null +++ b/docs/inventories/GeometricOptimizers.toml @@ -0,0 +1,955 @@ +# DocInventory version 1 +project = "GeometricOptimizers.jl" +version = "0.3.1" + +[[jl.constant]] +name = "GeometricOptimizers.CURVATURE_TOLERANCE" +uri = "index.html#$" +[[jl.constant]] +name = "GeometricOptimizers.DEFAULT_STEP_CEILING" +uri = "index.html#$" + +[[jl.function]] +name = "GeometricOptimizers.geodesic" +uri = "index.html#$" +[[jl.function]] +name = "GeometricOptimizers.manifold_type" +uri = "index.html#$" +[[jl.function]] +name = "GeometricOptimizers.𝔄exp" +uri = "index.html#GeometricOptimizers.%F0%9D%94%84exp" + +[[jl.method]] +name = 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+[[std.label]] +dispname = "The threshold θ needs no tuning" +name = "The-threshold-θ-needs-no-tuning" +uri = "retractions.html#The-threshold-%CE%B8-needs-no-tuning" +[[std.label]] +dispname = "Time-Dependent Vector Fields" +name = "Time-Dependent-Vector-Fields" +uri = "manifolds/existence_and_uniqueness_theorem.html#$" +[[std.label]] +dispname = "Two unrelated decays" +name = "Two-unrelated-decays" +uri = "weight_decay.html#$" +[[std.label]] +dispname = "Using them" +name = "Using-them" +uri = "retractions.html#$" +[[std.label]] +dispname = "Vector Fields" +name = "Vector-Fields" +uri = "manifolds/manifolds.html#$" +[[std.label]] +dispname = "Weight Decay on Manifolds" +name = "Weight-Decay-on-Manifolds" +uri = "weight_decay.html#$" +[[std.label]] +dispname = "Weights on Manifolds" +name = "Weights-on-Manifolds" +uri = "optimizer_methods.html#$" +[[std.label]] +dispname = "What a retraction is" +name = "What-a-retraction-is" +uri = "retractions.html#$" +[[std.label]] +dispname = "What the argument does and does not depend on" +name = "What-the-argument-does-and-does-not-depend-on" +uri = "weight_decay.html#$" +[[std.label]] +dispname = "What they cost" +name = "What-they-cost" +uri = "retractions.html#$" +[[std.label]] +dispname = "What they cost and how accurate they are" +name = "What-they-cost-and-how-accurate-they-are" +uri = "retractions.html#$" +[[std.label]] +dispname = "Where GeometricMachineLearning's method belongs" +name = "Where-GeometricMachineLearning's-method-belongs" +uri = "weight_decay.html#Where-GeometricMachineLearning%27s-method-belongs" +[[std.label]] +dispname = "Where a retraction sits in the algorithm" +name = "Where-a-retraction-sits-in-the-algorithm" +uri = "retractions.html#$" +[[std.label]] +dispname = "Which line search" +name = "Which-line-search" +uri = "weight_decay.html#$" +[[std.label]] +dispname = "Why the decay vanishes" +name = "Why-the-decay-vanishes" +uri = "weight_decay.html#$" +[[std.label]] +dispname = "Why the name is not AdamW" +name = "Why-the-name-is-not-AdamW" +uri = "weight_decay.html#$" +[[std.label]] +dispname = "Why the storage matters here" +name = "Why-the-storage-matters-here" +uri = "special_matrices.html#$" diff --git a/docs/make.jl b/docs/make.jl index 0381fd944..d0893f1ea 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -3,11 +3,34 @@ using HDF5 using AbstractNeuralNetworks using Documenter using DocumenterCitations +using DocumenterInterLinks using Markdown using Bibliography using LaTeXStrings # using Weave +# The manifold, special-matrix and optimizer chapters moved to `GeometricOptimizers` along with the +# types they describe (see the changelog), and the chapters that stayed refer to them constantly. +# This is what keeps those references *references* rather than prose. Closes issue C3, which asked +# for the same thing for `𝔄`, `cayley` and `update!`. +# +# The inventory is given as a *committed file* rather than as a URL. A URL is fetched and takes +# precedence over the fallback file, and the anchors these references need — the moved chapters — +# only exist in GeometricOptimizers' published inventory once its 0.4.0 docs have deployed. Reading +# the committed copy makes this build independent of that ordering, and of the network. Regenerate it +# after an upstream docs change with +# +# julia --project=docs -e 'using DocInventories; save("docs/inventories/GeometricOptimizers.toml", +# Inventory("../GeometricOptimizers/docs/build/objects.inv"; +# root_url = "https://juliagni.github.io/GeometricOptimizers.jl/stable/"))' +# +links = InterLinks( + "GeometricOptimizers" => ( + "https://juliagni.github.io/GeometricOptimizers.jl/stable/", + joinpath(@__DIR__, "inventories", "GeometricOptimizers.toml") + ), +) + bib = CitationBibliography(joinpath(@__DIR__, "src", "GeometricMachineLearning.bib")) sort_bibliography!(bib.entries, :nyt) # name-year-title @@ -123,19 +146,12 @@ _introduction = output_type == :html ? ("HOME" => "index.md") : ("HOME" => "introduction.md"] ) -_manifolds = "Manifolds" => [ - "Concepts from General Topology" => "manifolds/basic_topology.md", - "Metric and Vector Spaces" => "manifolds/metric_and_vector_spaces.md", - "Foundations of Differential Manifolds" => "manifolds/inverse_function_theorem.md", - "General Theory on Manifolds" => "manifolds/manifolds.md", - "Differential Equations and the EAU theorem" => "manifolds/existence_and_uniqueness_theorem.md", - "Riemannian Manifolds" => "manifolds/riemannian_manifolds.md", - "Homogeneous Spaces" => "manifolds/homogeneous_spaces.md", - ] +# The `Manifolds` chapter — general topology through homogeneous spaces — moved to +# `GeometricOptimizers` with the manifold types it describes, as did `Symmetric and Skew-Symmetric +# Matrices` and `Global Tangent Spaces` below and the whole `Optimizer` chapter. They are linked +# into from here through `InterLinks`. _special_arrays = "Special Arrays and AD" => [ - "Symmetric and Skew-Symmetric Matrices" => "arrays/skew_symmetric_matrix.md", - "Global Tangent Spaces" => "arrays/global_tangent_spaces.md", "Tensors" => "arrays/tensors.md", "Pullbacks" => "pullbacks/computation_of_pullbacks.md", ] @@ -146,13 +162,9 @@ _structure_preservation = "Structure-Preservation" => [ "Structure-Preserving Neural Networks" => "structure_preservation/structure_preserving_neural_networks.md", ] -optimizer_name = output_type == :html ? "Optimizer" : "Optimizer Framework" -_optimizers = optimizer_name => [ - "Optimizers" => "optimizers/optimizer_framework.md", - "Retractions" => "optimizers/manifold_related/retractions.md", - "Parallel Transport" => "optimizers/manifold_related/parallel_transport.md", - "Optimizer Methods" => "optimizers/optimizer_methods.md", - ] +# What is left of the optimizer chapter: the framework belongs to `GeometricOptimizers`, and this +# page covers the part that is about neural networks — the parameter tree and the training loop. +_optimizers = "Optimizer" => "optimizers/optimizer.md" _special_layers = "Special Neural Network Layers" => [ "Sympnet Layers" => "layers/sympnet_gradient.md", @@ -208,7 +220,6 @@ _index_of_docstrings = "Index of Docstrings" => "docstring_index.md" _html_pages = [ _introduction, - _manifolds, _special_arrays, _structure_preservation, _optimizers, @@ -288,17 +299,17 @@ end _latex_pages = [ _introduction, + # The `Manifolds` chapter this part opened with, and the `Optimizers` part that followed it, + # are `GeometricOptimizers`' documentation now. The book therefore starts from the geometric + # structure and takes the manifold optimizers as given; see the changelog. "Background" => [ - "Manifolds" => vcat(reduce_to_second_factors(_manifolds), - value_for_key(_special_arrays, "Global Tangent Spaces"), - ), "Geometric Structure" => reduce_to_second_factors(_structure_preservation), "Reduced Order Modeling" => reduce_to_second_factors(_reduced_order_modeling), ], - "Optimizers" => [ "General Framework for Manifold Optimization" => value_for_key(_optimizers, "Optimizers", "Retractions", "Parallel Transport"), - "Optimizer Methods" => - value_for_key(_optimizers, "Optimizer Methods") - ], + # One page, but it still has to be a chapter of *pairs* like the others: `index_latex_pages` + # below flattens these values and `docstring_index.md` builds a `Dict` from the result, so a + # chapter contributing a bare string breaks that `Dict`. + "Optimizer" => ["Optimizer" => reduce_to_second_factors(_optimizers)], "Special Neural Network Layers and Architectures" => [ "Layers" => reduce_to_second_factors(_special_layers), "Architectures" => reduce_to_second_factors(_architectures) @@ -319,9 +330,8 @@ _latex_pages = [ _index_of_docstrings, "Appendix" => [ "Data Loader" => reduce_to_second_factors(_data_loader), - "Special Arrays, Tensors and Pullbacks" => - value_for_key(_special_arrays, "Symmetric and Skew-Symmetric Matrices", - "Tensors", + "Tensors and Pullbacks" => + value_for_key(_special_arrays, "Tensors", "Pullbacks"), # we include the last tutorial here "Customizing Training" => value_for_key(_tutorials, "Adjusting the Loss Function"), @@ -335,7 +345,7 @@ filter!(key -> (key ≠ "HOME") & (key ≠ "Index of Docstrings") & (key ≠ "Re index_latex_pages = vcat([Dict(_latex_pages)[key] for key in _keys]...) makedocs(; - plugins = [bib], + plugins = [bib, links], # `GeometricOptimizers` is deliberately *not* listed. `@docs` filters candidate docstrings by # the module they were written in (`d.data[:module]`), not by the module of the binding, so the # `geodesic`/`cayley` methods GML defines on GeometricOptimizers' functions are found from here diff --git a/docs/src/architectures/linear_symplectic_transformer.md b/docs/src/architectures/linear_symplectic_transformer.md index 9bc972697..b5f8ad34f 100644 --- a/docs/src/architectures/linear_symplectic_transformer.md +++ b/docs/src/architectures/linear_symplectic_transformer.md @@ -17,7 +17,7 @@ The [standard transformer](@ref "Standard Transformer"), the [volume-preserving ![Two trajectories of a parameter-dependent ODE with the same initial condition.](../tikz/multiple_parameters_dark.png) -The trajectories come from a parameter-dependent [ODE](@ref "The Existence-And-Uniqueness Theorem") in two dimensions. As initial condition we take ``A\in\mathbb{R}^2`` and we look at two different parameter instances: ``\mu_1`` and ``\mu_2``. As we can see the curves ``\tilde{z}_{\mu_1}`` and ``\tilde{z}_{\mu_2}`` both start out at ``A,`` then go into different directions but cross again at ``D.`` If we used a standard feedforward neural network to treat this system it would not be able to resolve those training data as the information would be ambiguous at points ``A`` and ``D,`` i.e. the network would not know what it should predict. If we however consider the information coming from points three points, either ``(A, B, D)`` or ``(A, C, D),`` then the network can learn to predict the next time step. We will elaborate more on this in the [tutorial section](@ref "Comparing Different `VolumePreservingAttention` Mechanisms"). +The trajectories come from a parameter-dependent [ODE](@extref GeometricOptimizers The-Existence-And-Uniqueness-Theorem) in two dimensions. As initial condition we take ``A\in\mathbb{R}^2`` and we look at two different parameter instances: ``\mu_1`` and ``\mu_2``. As we can see the curves ``\tilde{z}_{\mu_1}`` and ``\tilde{z}_{\mu_2}`` both start out at ``A,`` then go into different directions but cross again at ``D.`` If we used a standard feedforward neural network to treat this system it would not be able to resolve those training data as the information would be ambiguous at points ``A`` and ``D,`` i.e. the network would not know what it should predict. If we however consider the information coming from points three points, either ``(A, B, D)`` or ``(A, C, D),`` then the network can learn to predict the next time step. We will elaborate more on this in the [tutorial section](@ref "Comparing Different `VolumePreservingAttention` Mechanisms"). ## Library Functions diff --git a/docs/src/architectures/neural_network_integrators.md b/docs/src/architectures/neural_network_integrators.md index 790e376af..44d690c7c 100644 --- a/docs/src/architectures/neural_network_integrators.md +++ b/docs/src/architectures/neural_network_integrators.md @@ -2,7 +2,7 @@ In `GeometricMachineLearning` we can divide most neural network architectures (that are used for applications to physical systems) into two categories: autoencoders and integrators. This is also closely related to the application of reduced order modeling where *autoencoders are used in the offline phase* and *integrators are used in the online phase*. -The term *integrator* in its most general form refers to an approximation of the [flow of an ODE](@ref "The Existence-And-Uniqueness Theorem") by a numerical scheme. Traditionally, for so called *one-step methods*, these numerical schemes are constructed by defining certain relationships between a known time step ``z^{(t)}`` and a future unknown one ``z^{(t+1)}`` [hairer2006geometric, leimkuhler2004simulating](@cite): +The term *integrator* in its most general form refers to an approximation of the [flow of an ODE](@extref GeometricOptimizers The-Existence-And-Uniqueness-Theorem) by a numerical scheme. Traditionally, for so called *one-step methods*, these numerical schemes are constructed by defining certain relationships between a known time step ``z^{(t)}`` and a future unknown one ``z^{(t+1)}`` [hairer2006geometric, leimkuhler2004simulating](@cite): ```math f(z^{(t)}, z^{(t+1)}) = 0. diff --git a/docs/src/architectures/symplectic_autoencoder.md b/docs/src/architectures/symplectic_autoencoder.md index 795680202..f57dfa4b6 100644 --- a/docs/src/architectures/symplectic_autoencoder.md +++ b/docs/src/architectures/symplectic_autoencoder.md @@ -27,7 +27,7 @@ where ``A_i^{(+)} = A_i`` if ``d_{i+1} > d_i`` and ``A_i^{(+)} = A_i^+`` if ``d_ \end{aligned} ``` -so the symplectic inverse is equivalent to a matrix transpose in this case. In the symplectic autoencoder we use SympNets as a form of *symplectic preprocessing* before the linear symplectic reduction (i.e. the PSD layer) is employed. The resulting neural network has some of its weights on manifolds, which is why we cannot use standard neural network optimizers, but have to resort to [manifold optimizers](@ref "Generalization to Homogeneous Spaces"). Note that manifold optimization is not necessary for the weights corresponding to the SympNet layers, these are still updated with standard neural network optimizers during training. Also note that SympNets are nonlinear and preserve symplecticity, but they cannot change the dimension of a system while PSD layers can change the dimension of a system and preserve symplecticity, but are strictly linear. Symplectic autoencoders have all three properties: they preserve symplecticity, can change dimension and are nonlinear mappings. We can visualize this in a Venn diagram: +so the symplectic inverse is equivalent to a matrix transpose in this case. In the symplectic autoencoder we use SympNets as a form of *symplectic preprocessing* before the linear symplectic reduction (i.e. the PSD layer) is employed. The resulting neural network has some of its weights on manifolds, which is why we cannot use standard neural network optimizers, but have to resort to [manifold optimizers](@extref GeometricOptimizers Generalization-to-Homogeneous-Spaces). Note that manifold optimization is not necessary for the weights corresponding to the SympNet layers, these are still updated with standard neural network optimizers during training. Also note that SympNets are nonlinear and preserve symplecticity, but they cannot change the dimension of a system while PSD layers can change the dimension of a system and preserve symplecticity, but are strictly linear. Symplectic autoencoders have all three properties: they preserve symplecticity, can change dimension and are nonlinear mappings. We can visualize this in a Venn diagram: ![Venn diagram visualizing that a symplectic autoencoder (SAE) is symplectic, can change dimension and is nonlinear.](../tikz/sae_venn_light.png) ![Venn diagram visualizing that a symplectic autoencoder (SAE) is symplectic, can change dimension and is nonlinear.](../tikz/sae_venn_dark.png) diff --git a/docs/src/architectures/sympnet.md b/docs/src/architectures/sympnet.md index 950155a69..72cea0d3c 100644 --- a/docs/src/architectures/sympnet.md +++ b/docs/src/architectures/sympnet.md @@ -153,7 +153,7 @@ with ``|\alpha| = \alpha_1 +...+ \alpha_{2d}``. We impose the following conditio Main.definition(raw"``\sigma`` is **``r``-finite** if ``\sigma\in C^r(\mathbb{R},\mathbb{R})`` and ``\int |D^r\sigma(x)|dx <\infty``.") ``` -We further consider the topology on ``C^r(U, \mathbb{R}^d)`` induced by ``||\cdot ||_{C^r(\cdot, \mathbb{R}^d)}`` and the associated notion of [denseness](@ref "Basic Concepts from General Topology"): +We further consider the topology on ``C^r(U, \mathbb{R}^d)`` induced by ``||\cdot ||_{C^r(\cdot, \mathbb{R}^d)}`` and the associated notion of [denseness](@extref GeometricOptimizers Basic-Concepts-from-General-Topology): ```@eval Main.definition(raw"Let ``m,d,r\in \mathbb{N}`` with ``m,d>0`` be given, ``U`` an open subset of ``\mathbb{R}^m``, and ``I,J\subset C^r(U,\mathbb{R}^d)``. We say ``J`` is **``r``-uniformly dense on compacta in ``I``** if ``J \subset I`` and for any ``f\in I``, ``\epsilon>0``, and any compact ``K\subset U``, there exists ``g\in J`` such that ``||f-g||_{C^r(K,\mathbb{R}^{d})} < \epsilon``.") @@ -183,7 +183,7 @@ There are many ``r``-finite activation functions commonly used in neural network - The sigmoid activation function: ``\sigma(x) = {1} / (1+e^{-x})``, - The hyperbolic tangent function: ``\tanh(x) = (e^x-e^{-x}) / (e^x+e^{-x})``. -The universal approximation theorems state that we can, in principle, get arbitrarily close to any symplectomorphism defined on ``\mathbb{R}^{2d}``. But this does not tell us anything about how to optimize the network. This is can be done with any common [neural network optimizer](@ref "Neural Network Optimizers") and these neural network optimizers always rely on a corresponding loss function. +The universal approximation theorems state that we can, in principle, get arbitrarily close to any symplectomorphism defined on ``\mathbb{R}^{2d}``. But this does not tell us anything about how to optimize the network. This is can be done with any common [neural network optimizer](@extref GeometricOptimizers The-optimizer-framework,-step-by-step) and these neural network optimizers always rely on a corresponding loss function. ## Loss function diff --git a/docs/src/arrays/global_tangent_spaces.md b/docs/src/arrays/global_tangent_spaces.md deleted file mode 100644 index 1cccadbe9..000000000 --- a/docs/src/arrays/global_tangent_spaces.md +++ /dev/null @@ -1,288 +0,0 @@ -# Global Tangent Spaces - -In `GeometricMachineLearning` standard neural network optimizers are generalized to [homogeneous spaces](@ref "Homogeneous Spaces") by leveraging the special structure of the tangent spaces of this class of manifolds. When we introduced homogeneous spaces we already talked about that every tangent space to a homogeneous space ``T_Y\mathcal{M}`` is of the form: - -```math - T_Y\mathcal{M} = \mathfrak{g} \cdot Y := \{AY: A\in{}\mathfrak{g}\}. -``` - -We then have a decomposition of ``\mathfrak{g}`` into a vertical part ``\mathfrak{g}^{\mathrm{ver}, Y}`` and a horizontal part ``\mathfrak{g}^{\mathrm{hor}, Y}`` and the horizontal part is isomorphic to ``T_Y\mathcal{M}`` via: - -```math - \mathfrak{g}^{\mathrm{hor}, Y} = \{\Omega(\Delta): \Delta\in{}T_Y\mathcal{M} \}. -``` - -We now identify a special element ``E \in \mathcal{M}`` and designate the horizontal component ``\mathfrak{g}^{\mathrm{hor}, E}`` as our *global tangent space*. We will refer to this global tangent space by ``\mathfrak{g}^\mathrm{hor}``. We can now find a transformation from any ``\mathfrak{g}^{\mathrm{hor}, Y}`` to ``\mathfrak{g}^\mathrm{hor}`` and vice-versa (these spaces are isomorphic). - -```@eval -Main.theorem(raw"Let ``A\in{}G`` an element such that ``AE = Y``. Then we have -" * Main.indentation * raw"```math -" * Main.indentation * raw"A^{-1}\cdot\mathfrak{g}^{\mathrm{hor},Y}\cdot{}A = \mathfrak{g}^\mathrm{hor}, -" * Main.indentation * raw"``` -" * Main.indentation * raw"i.e. for every element ``B\in\mathfrak{g}^\mathrm{hor}`` we can find a ``B^Y \in \mathfrak{g}^{\mathrm{hor},Y}`` s.t. ``B = A^{-1}B^YA`` (and vice-versa).") -``` - -```@eval -Main.proof(raw"We first show that for every ``B^Y\in\mathfrak{g}^{\mathrm{hor},Y}`` the element ``A^{-1}B^YA`` is in ``\mathfrak{g}^{\mathrm{hor}}``. First note that ``A^{-1}B^YA\in\mathfrak{g}`` by a fundamental theorem of Lie group theory (closedness of the Lie algebra under adjoint action). Now assume that ``A^{-1}B^YA`` is not fully contained in ``\mathfrak{g}^\mathrm{hor}``, i.e. it also has a vertical component. So we would lose information when performing ``A^{-1}B^YA \mapsto A^{-1}B^YAE = A^{-1}B^YY``, but this contradicts the fact that ``B^Y\in\mathfrak{g}^{\mathrm{hor},Y}.`` We now have to proof that for every ``B\in\mathfrak{g}^\mathrm{hor}`` we can find an element in ``\mathfrak{g}^{\mathrm{hor}, Y}`` such that this element is mapped to ``B``. By a argument similar to the one above we can show that ``ABA^{-1}\in\mathfrak{g}^\mathrm{hor, Y}`` and this element maps to ``B``. Proofing that the map is injective is now trivial.") -``` - -We should note that we have written all Lie group and Lie algebra actions as simple matrix multiplications, like ``AE = Y``. For some Lie groups and Lie algebras, as the Lie group of isomorphisms on some domain ``\mathcal{D}``, this notation may not be appropriate [holm2009geometric](@cite). These Lie groups are however not relevant for what we use in `GeometricMachineLearning` and we will stick to regular matrix notation. - -## Global Sections - -Note that the theorem above requires us to find an element ``A\in{}G`` such that ``AE = Y``. We will call such a mapping ``\lambda:\mathcal{M}\to{}G`` a *global section*[^1]. - -[^1]: Global sections are also crucial for [parallel transport](@ref "Parallel Transport") in `GeometricMachineLearning`. A global section is first updated, i.e. ``\Lambda^{(t)} \gets \mathrm{update}(\Lambda^{(t-1)});`` and on the basis of this we then update the element of the manifold ``Y\in\mathcal{M}`` and the tangent vector ``\Delta\in{}T\mathcal{M}``. - -```@eval -Main.definition(raw"We call a mapping ``\lambda`` from a homogeneous space ``\mathcal{M}`` to its associated Lie group ``G`` a **global section** if ``\forall{}Y\in\mathcal{M}`` it satisfies: -" * Main.indentation * raw"```math -" * Main.indentation * raw"\lambda(Y)E = Y, -" * Main.indentation * raw"``` -" * Main.indentation * raw"where ``E`` is the distinct element of the homogeneous space.") -``` - -Note that in general global sections are not unique because the rank of ``G`` is in general greater than that of ``\mathcal{M}``. We give an example of how to construct such a global section for the Stiefel and the Grassmann manifolds below. - -## The Global Tangent Space for the Stiefel Manifold - -We now discuss the specific form of the global tangent space for the [Stiefel manifold](@ref "The Stiefel Manifold"). We pick as distinct element ``E`` (which build by calling [`StiefelProjection`](@ref)): - -```math -E = \begin{bmatrix} -\mathbb{I}_n \\ -\mathbb{O} -\end{bmatrix}\in{}St(n, N). -``` - -Based on this, elements of the vector space ``\mathfrak{g}^{\mathrm{hor}, E} =: \mathfrak{g}^{\mathrm{hor}}`` are: - -```math -\bar{B} = \begin{pmatrix} -A & B^T \\ B & \mathbb{O} -\end{pmatrix}, -``` - -where ``A`` is a skew-symmetric matrix of size ``n\times{}n`` and ``B`` is an arbitrary matrix of size ``(N - n)\times{}n``. Arrays of type ``\mathfrak{g}^{\mathrm{hor}, E} \equiv \mathfrak{g}^\mathrm{hor}`` are implemented in `GeometricMachineLearning` under the name [`StiefelLieAlgHorMatrix`](@ref). - -We can call this with e.g. a skew-symmetric matrix ``A`` and an arbitrary matrix ``B``: - -```@example call_stiefel_lie_alg_hor_matrix_1 -using GeometricMachineLearning # hide - -N, n = 5, 2 - -A = rand(SkewSymMatrix, n) -``` - -```@example call_stiefel_lie_alg_hor_matrix_1 -B = rand(N - n, n) -``` - -The constructor is then called as follows: - -```@example call_stiefel_lie_alg_hor_matrix_1 -B̄ = StiefelLieAlgHorMatrix(A, B, N, n) -``` - -We can also call it with a matrix of shape ``N\times{}N``: - -```@example call_stiefel_lie_alg_hor_matrix_1 -B̄₂ = Matrix(B̄) # note that this does not have any special structure - -StiefelLieAlgHorMatrix(B̄₂, n) -``` - -Or we can call it on ``T_E\mathcal{M}\subset\mathbb{R}^{N\times{}n},`` i.e. a matrix of shape ``N\times{}n``: - -```@example call_stiefel_lie_alg_hor_matrix_1 -E = StiefelProjection(N, n) -``` - -```@example call_stiefel_lie_alg_hor_matrix_1 -B̄₃ = B̄ * E - -StiefelLieAlgHorMatrix(B̄₃, n) -``` - -We now demonstrate how to map from an element of ``\mathfrak{g}^{\mathrm{hor}, Y}`` to an element of ``\mathfrak{g}^\mathrm{hor}``: - -```@example global_section -using GeometricMachineLearning # hide -using GeometricMachineLearning: Ω - -N, n = 5, 2 # hide -Y = rand(StiefelManifold, N, n) -Δ = rgrad(Y, rand(N, n)) -ΩΔ = Ω(Y, Δ) -λY = GlobalSection(Y) - -λY_mat = Matrix(λY) - -round.(λY_mat' * ΩΔ * λY_mat; digits = 3) -``` - -Performing this computation directly is computationally very inefficient however and the user is strongly discouraged to call `Matrix` on an instance of `GlobalSection`. The better option is calling `global_rep`: - -```@example global_section -using GeometricMachineLearning: _round # hide - -_round(global_rep(λY, Δ); digits = 3) -``` - -Internally `GlobalSection` calls the function [`GeometricMachineLearning.global_section`](@ref) which does the following for the Stiefel manifold: - -```julia -A = randn(N, N - n) # or the gpu equivalent -A = A - Y * (Y' * A) -Y⟂ = qr(A).Q[1:N, 1:(N - n)] -``` - -So we draw ``(N - n)`` new columns randomly, subtract the part that is spanned by the columns of ``Y`` and then perform a ``QR`` composition on the resulting matrix. The ``Q`` part of the decomposition is a matrix of ``(N - n)`` columns that is orthogonal to ``Y`` and is typically referred to as ``Y_\perp`` [absil2004riemannian, absil2008optimization, bendokat2020grassmann](@cite). We can easily check that this ``Y_\perp`` is indeed orthogonal to ``Y``. - -```@eval -Main.theorem(raw"The matrix ``Y_\perp`` constructed with the algorithm above satisfies -" * Main.indentation * raw"```math -" * Main.indentation * raw"Y^TY_\perp = \mathbb{O}_{n\times{}n}, -" * Main.indentation * raw"``` -" * Main.indentation * raw"and -" * Main.indentation * raw"```math -" * Main.indentation * raw"(Y_\perp)^TY_\perp = \mathbb{I}_n, -" * Main.indentation * raw"``` -" * Main.indentation * raw"i.e. all the columns in the big matrix ``[Y, Y_\perp]\in\mathbb{R}^{N\times{}N}`` are mutually orthonormal and it therefore is an element of ``SO(N)``.") -``` - -```@eval -Main.proof(raw"The second property is trivially satisfied because the ``Q`` component of a ``QR`` decomposition is an orthogonal matrix. For the first property note that ``Y^TQR = \mathbb{O}`` is zero because we have subtracted the ``Y`` component from the matrix ``QR``. The matrix ``R\in\mathbb{R}^{N\times{}(N-n)}`` further has the property ``[R]_{ij} = 0`` for ``i > j`` and we have that -" * Main.indentation * raw"```math -" * Main.indentation * raw"(Y^TQ)R = [r_{11}(Y^TQ)_{1\bullet}, r_{12}(Y^TQ)_{1\bullet} + r_{22}(Y^TQ)_{2\bullet}, \ldots, \sum_{i=1}^{N-n}r_{i(N-n)}(Y^TQ)_{i\bullet}]. -" * Main.indentation * raw"``` -" * Main.indentation * raw"Now all the coefficients ``r_{ii}`` are non-zero because the matrix we performed the ``QR`` decomposition on has full rank and we can see that if ``(Y^TQ)R`` is zero ``Y^TQ`` also has to be zero.") -``` - -The function `global_rep` furthermore makes use of the following: - -```math - \mathtt{global\_rep}(Y) = \lambda(Y)^T\Omega(Y,\Delta)\lambda(Y) = EY^T\Delta{}E^T + \begin{bmatrix} \mathbb{O} \\ \bar{\lambda}^T\Delta{}E^T \end{bmatrix} - \begin{bmatrix} \mathbb{O} & E\Delta^T\bar{\lambda} \end{bmatrix}, -``` -where ``\lambda(Y) = [Y, \bar{\lambda}].`` - -```@eval -Main.proof(raw"We derive the expression above: -" * Main.indentation * raw"```math -" * Main.indentation * raw"\begin{aligned} -" * Main.indentation * raw"\lambda(Y)^T\Omega(Y,\Delta)\lambda(Y) & = \lambda(Y)^T[(\mathbb{I} - \frac{1}{2}YY^T)\Delta{}Y^T - Y\Delta^T(\mathbb{I} - \frac{1}{2}YY^T)]\lambda(Y) \\ -" * Main.indentation * raw" & = \lambda(Y)^T[(\mathbb{I} - \frac{1}{2}YY^T)\Delta{}E^T - Y\Delta^T(\lambda(Y) - \frac{1}{2}YE^T)] \\ -" * Main.indentation * raw" & = \lambda(Y)^T\Delta{}E^T - \frac{1}{2}EY^T\Delta{}E^T - E\Delta^T\lambda(Y) + \frac{1}{2}E\Delta^TYE^T \\ -" * Main.indentation * raw" & = \begin{bmatrix} Y^T\Delta{}E^T \\ \bar{\lambda}\Delta{}E^T \end{bmatrix} - \frac{1}{2}EY^T\Delta{}E - \begin{bmatrix} E\Delta^TY & E\Delta^T\bar{\lambda} \end{bmatrix} + \frac{1}{2}E\Delta^TYE^T \\ -" * Main.indentation * raw" & = \begin{bmatrix} Y^T\Delta{}E^T \\ \bar{\lambda}\Delta{}E^T \end{bmatrix} + E\Delta^TYE^T - \begin{bmatrix}E\Delta^TY & E\Delta^T\bar{\lambda} \end{bmatrix} \\ -" * Main.indentation * raw" & = EY^T\Delta{}E^T + E\Delta^TYE^T - E\Delta^TYE^T + \begin{bmatrix} \mathbb{O} \\ \bar{\lambda}\Delta{}E^T \end{bmatrix} - \begin{bmatrix} \mathbb{O} & E\Delta^T\bar{\lambda} \end{bmatrix} \\ -" * Main.indentation * raw" & = EY^T\Delta{}E^T + \begin{bmatrix} \mathbb{O}_{n\times{}N} \\ \bar{\lambda}\Delta{}E^T \end{bmatrix} - \begin{bmatrix} \mathbb{O}_{N\times{}n} & E\Delta^T\bar{\lambda} \end{bmatrix}, -" * Main.indentation * raw"\end{aligned} -" * Main.indentation * raw"``` -" * Main.indentation * raw"which proofs our assertion.") -``` - -This expression of `global_rep` means we only need ``Y^T\Delta`` and ``\bar{\lambda}^T\Delta`` and this is what is used internally. - -We now discuss the global tangent space for the Grassmann manifold. This is similar to the Stiefel case. - -## Global Tangent Space for the Grassmann Manifold - -In the case of the Grassmann manifold we construct the global tangent space with respect to the distinct element ``\mathcal{E}=\mathrm{span}(E)\in{}Gr(n,N)``, where ``E`` is again the same matrix. - -The tangent tangent space ``T_\mathcal{E}Gr(n,N)`` can be represented through matrices: - -```math -\begin{pmatrix} - 0 & \cdots & 0 \\ - \cdots & \cdots & \cdots \\ - 0 & \cdots & 0 \\ - b_{11} & \cdots & b_{1n} \\ - \cdots & \cdots & \cdots \\ - b_{(N-n)1} & \cdots & b_{(N-n)n} -\end{pmatrix}. -``` - -This representation is based on the identification ``T_\mathcal{E}Gr(n,N)\to{}T_E\mathcal{S}_E`` that was discussed in the section on the [Grassmann manifold](@ref "The Grassmann Manifold")[^2]. We use the following notation: - -[^2]: We derived the following expression for the Riemannian gradient of the Grassmann manifold: ``\mathrm{grad}_\mathcal{Y}^{Gr}L = \nabla_Y{}L - YY^T\nabla_YL``. The tangent space to the element ``\mathcal{E}`` can thus be written as ``\bar{B} - EE^T\bar{B}`` where ``B\in\mathbb{R}^{N\times{}n}`` and the matrices in this tangent space have the desired form. - -```math -\mathfrak{g}^\mathrm{hor} = \mathfrak{g}^{\mathrm{hor},\mathcal{E}} = \left\{\begin{pmatrix} 0 & -B^T \\ B & 0 \end{pmatrix}: \text{$B\in\mathbb{R}^{(N-n)\times{}n}$ is arbitrary}\right\}. -``` - -This is equivalent to the horizontal component of ``\mathfrak{g}`` for the Stiefel manifold for the case when ``A`` is zero. This is a reflection of the rotational invariance of the Grassmann manifold: the skew-symmetric matrices ``A`` are connected to the group of rotations ``O(n)`` which is factored out in the Grassmann manifold ``Gr(n,N)\simeq{}St(n,N)/O(n)``. In `GeometricMachineLearning` we thus treat the Grassmann manifold as being embedded in the Stiefel manifold. In [bendokat2020grassmann](@cite) viewing the Grassmann manifold as a quotient space of the Stiefel manifold is important for "feasibility" in "practical computations". - -## Library Functions - -```@docs -GeometricMachineLearning.AbstractLieAlgHorMatrix -StiefelLieAlgHorMatrix -StiefelLieAlgHorMatrix(::AbstractMatrix, ::Int) -GrassmannLieAlgHorMatrix -GrassmannLieAlgHorMatrix(::AbstractMatrix, ::Int) -vec(::StiefelLieAlgHorMatrix) -GeometricMachineLearning.global_section(::StiefelManifold{T}) where T -GeometricMachineLearning.global_section(::GrassmannManifold{T}) where T -``` - -`GlobalSection` itself, together with `Matrix(::GlobalSection)`, `apply_section`, -`apply_section!`, `λY * Y` and `global_rep`, is provided by -[`GeometricOptimizers`](https://juliagni.github.io/GeometricOptimizers.jl/stable/) and documented in -its manual; `GeometricMachineLearning` re-exports these names and supplies the methods for its own -manifold types. - - -```@raw latex -\section*{Chapter Summary} - -In this chapter we discussed mathematical core concepts in this dissertation: aspects from general topology and analysis. We furthermore discussed Riemannian geometry and parallel transport which are crucial concepts for manifold optimization. Homogeneous spaces were introduced as an important subcategory of manifolds and we showed how to find a global tangent space representation for them. This was done by identifying a distinct element $E\in\mathcal{M}$ and then considering the action of the Lie algebra $\mathfrak{g}$ on this element. We have presented our basic application user interface (API) for the \texttt{Julia} package \texttt{GeometricMachineLearning}. This API will be extended in Part II with a general optimizer framework. -``` - -```@raw latex -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -absil2004riemannian -absil2008optimization -bendokat2020grassmann -brantner2023generalizing -frankel2011geometry -``` - -```@raw latex -\end{comment} -``` - -```@raw html - -``` \ No newline at end of file diff --git a/docs/src/arrays/skew_symmetric_matrix.md b/docs/src/arrays/skew_symmetric_matrix.md deleted file mode 100644 index 5b425fb0c..000000000 --- a/docs/src/arrays/skew_symmetric_matrix.md +++ /dev/null @@ -1,166 +0,0 @@ -```@raw latex -\texttt{GeometricMachineLearning} has custom versions of matrices such as the symmetric and the skew-symmetric matrix implemented. These are important ingredients in e.g. SympNets and volume-preserving transformers and it is therefore important that those implementations also run efficiently on GPU. We also show how to build custom pullbacks for specific functions in \texttt{Julia}. -``` - -# Symmetric, Skew-Symmetric and Triangular Matrices. - -Among the special arrays implemented in `GeometricMachineLearning` [`SymmetricMatrix`](@ref), [`SkewSymMatrix`](@ref), [`UpperTriangular`](@ref) and [`LowerTriangular`](@ref) are the most common ones and similar implementations can also be found in other libraries; `LinearAlgebra.jl` has an implementation of a symmetric matrix called [`Symmetric`](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.Symmetric) for example. The versions of these matrices in `GeometricMachineLearning` are however more memory efficient as they only store as many parameters as are necessary, i.e. ``n(n+1)/2`` for the symmetric matrix and ``n(n-1)/2`` for the other three. In addition operations such as matrix and tensor multiplication are implemented for these matrices to work in parallel on GPU via [`GeometricMachineLearning.tensor_mat_mul`](@ref) for example. We here give an overview of *elementary* custom matrices that are implemented in `GeometricMachineLearning`. More *involved* matrices are the so-called [global tangent spaces](@ref "Global Tangent Spaces"). - -## Custom Matrices - -`GeometricMachineLearning` has two types of *triangular matrices*. The first one is [`UpperTriangular`](@ref): - -```math -U = \begin{pmatrix} - 0 & a_{12} & \cdots & a_{1n} \\ - 0 & \ddots & & a_{2n} \\ - \vdots & \ddots & \ddots & \vdots \\ - 0 & \cdots & 0 & 0 -\end{pmatrix}. -``` - -And the second one is [`LowerTriangular`](@ref): - -```math -L = \begin{pmatrix} - 0 & 0 & \cdots & 0 \\ - a_{21} & \ddots & & \vdots \\ - \vdots & \ddots & \ddots & \vdots \\ - a_{n1} & \cdots & a_{n(n-1)} & 0 -\end{pmatrix}. -``` - -An instance of [`SkewSymMatrix`](@ref) can be written as ``A = L - L^T`` or ``A = U^T - U``: - -```math -A = \begin{pmatrix} - 0 & - a_{21} & \cdots & - a_{n1} \\ - a_{21} & \ddots & & \vdots \\ - \vdots & \ddots & \ddots & \vdots \\ - a_{n1} & \cdots & a_{n(n-1)} & 0 -\end{pmatrix}. -``` - -And lastly a [`SymmetricMatrix`](@ref): - -```math -B = \begin{pmatrix} - a_{11} & a_{21} & \cdots & a_{n1} \\ - a_{21} & \ddots & & \vdots \\ - \vdots & \ddots & \ddots & \vdots \\ - a_{n1} & \cdots & a_{n(n-1)} & a_{nn} -\end{pmatrix}. -``` - -Note that any matrix ``M\in\mathbb{R}^{n\times{}n}`` can be written - -```math -M = \frac{1}{2}(M - M^T) + \frac{1}{2}(M + M^T), -``` -where the first part of this matrix is skew-symmetric and the second part is symmetric. This is also how the constructors for [`SkewSymMatrix`](@ref) and [`SymmetricMatrix`](@ref) are designed. Consider an arbitrary matrix: - -```@example sym_skew_sym_example -using GeometricMachineLearning # hide - -M = [1; 2; 3;; 4; 5; 6;; 7; 8; 9] -``` - -Calling [`SkewSymMatrix`](@ref) on ``M`` is equivalent to doing ``M \to \frac{1}{2}(M - M^T)``: - -```@example sym_skew_sym_example -A = SkewSymMatrix(M) -``` - -And calling [`SymmetricMatrix`](@ref) on ``M`` is equivalent to doing ``M \to \frac{1}{2}(M + M^T)``: - -```@example sym_skew_sym_example -B = SymmetricMatrix(M) -``` - -We can further confirm the identity above: - -```@example sym_skew_sym_example -@assert M ≈ A + B # hide -M ≈ A + B -``` - -Note that for [`LowerTriangular`](@ref) and [`UpperTriangular`](@ref) no projection step is involved, which means that if we start with a matrix of type `AbstractMatrix{Int64}` we will end up with a matrix that is also of type `AbstractMatrix{Int64}`. The type changes however when we call [`SkewSymMatrix`](@ref) and [`SymmetricMatrix`](@ref): - -```@example sym_skew_sym_example -@assert (typeof(A) <: AbstractMatrix{Int64}) == false # hide -@assert (typeof(B) <: AbstractMatrix{Int64}) == false # hide -(typeof(A) <: AbstractMatrix{Int64}, typeof(B) <: AbstractMatrix{Int64}) -``` - -For the triangular matrices: - -```@example sym_skew_sym_example -U = UpperTriangular(M) -L = LowerTriangular(M) -@assert (typeof(U) <: AbstractMatrix{Int64}) == true # hide -@assert (typeof(L) <: AbstractMatrix{Int64}) == true # hide -(typeof(U) <: AbstractMatrix{Int64}, typeof(L) <: AbstractMatrix{Int64}) -``` - -## How are Special Matrices Stored? - -The following image demonstrates how a skew-symmetric matrix is stored in `GeometricMachineLearning`: - -![The elements of a skew-symmetric matrix (and other special matrices) are stored as a vector. The elements of the big vector are the entries on the lower left of the matrix, stored row-wise.](../tikz/skew_sym_visualization_light.png) -![The elements of a skew-symmetric matrix (and other special matrices) are stored as a vector. The elements of the big vector are the entries on the lower left of the matrix, stored row-wise.](../tikz/skew_sym_visualization_dark.png) - -So what is stored internally is a vector of size ``n(n-1)/2`` for the skew-symmetric matrix and the triangular matrices, and a vector of size ``n(n+1)/2`` for the symmetric matrix. - -## Sample Random Matrices - -We can sample a random skew-symmetric matrix: - -```@example skew_sym -using GeometricMachineLearning # hide -import Random # hide -Random.seed!(123) # hide - -A = rand(SkewSymMatrix, 3) -``` - -and then access the vector: - -```@example skew_sym -A.S -``` - -This is equivalent to sampling a vector and then assigning a matrix[^1]: - -[^1]: We fixed the seed to the same value in both these examples. - -```@example skew_sym -using GeometricMachineLearning # hide -import Random # hide -Random.seed!(123) # hide - -S = rand(3 * (3 - 1) ÷ 2) -@assert A == SkewSymMatrix(S, 3) # hide -SkewSymMatrix(S, 3) -``` - -These special matrices are important for [SympNets](@ref "SympNet Architecture"), [volume-preserving transformers](@ref "Volume-Preserving Transformer") and [linear symplectic transformers](@ref "Linear Symplectic Transformer"). - -## Parallel Computation - -The functions [`GeometricMachineLearning.mat_tensor_mul`](@ref) and [`GeometricMachineLearning.tensor_mat_mul`](@ref) are also implemented for these matrices for efficient parallel computations. This is elaborated on when we take about [tensors](@ref "Tensors in `GeometricMachineLearning`"). - -## Library Functions - -```@docs -GeometricMachineLearning.AbstractTriangular -UpperTriangular -UpperTriangular(::AbstractMatrix) -LowerTriangular -LowerTriangular(::AbstractMatrix) -vec(::GeometricMachineLearning.AbstractTriangular) -SkewSymMatrix -SkewSymMatrix(::AbstractMatrix) -SymmetricMatrix -SymmetricMatrix(::AbstractMatrix) -vec(::SkewSymMatrix) -``` \ No newline at end of file diff --git a/docs/src/arrays/tensors.md b/docs/src/arrays/tensors.md index 077ddb0e2..a7e0d1eac 100644 --- a/docs/src/arrays/tensors.md +++ b/docs/src/arrays/tensors.md @@ -1,6 +1,6 @@ # Tensors in `GeometricMachineLearning` -We typically store training data as *tensors with three axes* in `GeometricMachineLearning`. This allows for a parallel computation of matrix products, also for the special arrays such as [`LowerTriangular`](@ref), [`UpperTriangular`](@ref), [`SymmetricMatrix`](@ref) and [`SkewSymMatrix`](@ref) and objects of [`Manifold`](@ref) type such as the [`StiefelManifold`](@ref). +We typically store training data as *tensors with three axes* in `GeometricMachineLearning`. This allows for a parallel computation of matrix products, also for the special arrays such as [`LowerTriangular`](@extref GeometricOptimizers GeometricOptimizers.LowerTriangular), [`UpperTriangular`](@extref GeometricOptimizers GeometricOptimizers.UpperTriangular), [`SymmetricMatrix`](@extref GeometricOptimizers GeometricOptimizers.SymmetricMatrix) and [`SkewSymMatrix`](@extref GeometricOptimizers GeometricOptimizers.SkewSymMatrix) and objects of [`Manifold`](@extref GeometricOptimizers GeometricOptimizers.Manifold) type such as the [`StiefelManifold`](@extref GeometricOptimizers GeometricOptimizers.StiefelManifold). ## Library Functions diff --git a/docs/src/docstring_index.md b/docs/src/docstring_index.md index 8c38d227c..274515361 100644 --- a/docs/src/docstring_index.md +++ b/docs/src/docstring_index.md @@ -4,12 +4,6 @@ \thispagestyle{empty} ``` -### Manifolds - -```@index -Pages = Dict(Main.index_latex_pages)["Manifolds"] -``` - ### Geometric Structure ```@index @@ -22,16 +16,10 @@ Pages = Dict(Main.index_latex_pages)["Geometric Structure"] Pages = Dict(Main.index_latex_pages)["Reduced Order Modeling"] ``` -### General Framework for Manifold Optimization - -```@index -Pages = Dict(Main.index_latex_pages)["General Framework for Manifold Optimization"] -``` - -### Optimizer Methods +### Optimizer ```@index -Pages = [Dict(Main.index_latex_pages)["Optimizer Methods"]] +Pages = [Dict(Main.index_latex_pages)["Optimizer"]] ``` ### Layers @@ -65,8 +53,8 @@ Pages = [Dict(Main.index_latex_pages)["Learning Nonlinear Spaces"]] Pages = Dict(Main.index_latex_pages)["Data Loader"] ``` -### Special Arrays, Tensors and Pullbacks +### Tensors and Pullbacks ```@index -Pages = Dict(Main.index_latex_pages)["Special Arrays, Tensors and Pullbacks"] +Pages = Dict(Main.index_latex_pages)["Tensors and Pullbacks"] ``` \ No newline at end of file diff --git a/docs/src/index.md b/docs/src/index.md index 8ef41c358..a789292cc 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -35,7 +35,7 @@ Existing architectures include: ## Manifolds -`GeometricMachineLearning` supports putting neural network weights on manifolds such as the [Stiefel manifold](@ref "The Stiefel Manifold") and the [Grassmann manifold](@ref "The Grassmann Manifold") and [Riemannian optimization](@ref "Riemannian Manifolds"). +`GeometricMachineLearning` supports putting neural network weights on manifolds such as the [Stiefel manifold](@extref GeometricOptimizers The-Stiefel-Manifold) and the [Grassmann manifold](@extref GeometricOptimizers The-Grassmann-Manifold) and [Riemannian optimization](@extref GeometricOptimizers Riemannian-Manifolds). ![Weights can be put on manifolds to achieve structure preservation or improved stability.](tikz/tangent_vector_light.png) ![Weights can be put on manifolds to achieve structure preservation or improved stability.](tikz/tangent_vector_dark.png) diff --git a/docs/src/introduction.md b/docs/src/introduction.md index ef356ad69..948c05e82 100644 --- a/docs/src/introduction.md +++ b/docs/src/introduction.md @@ -19,10 +19,10 @@ Closely linked to the research presented here is the development of a software p [^0]: This document was produced with `GeometricMachineLearning` `v0.3`. It may be that the interface will slightly change in future versions, but efforts will be made to keep these changes as small as possible. ```@docs; canonical = false -rgrad(::StiefelManifold, ::AbstractMatrix) +optimization_step! ``` -So the docstring shows the name of the type or method, in most cases how to call it and then gives some information explaining what it does and potentially hyperlinks to other similar docstrings ([`metric(::StiefelManifold, ::AbstractMatrix, ::AbstractMatrix)`](@ref) in this case); all of this information is indented by a tab. Docstrings may include other information under subheaders **Arguments** (showing the arguments the method can be supplied with), **Examples** (giving more detailed examples (including results) of how to use the method) and **Implementation** (giving details on how the method is implemented). When we reference a docstring it is always printed in blue (e.g. [`rgrad(::StiefelManifold, ::AbstractMatrix)`](@ref)), indicating a hyperlink. In addition there is an *index of docstrings* showing all docstrings in chronological order with the associated page number. +So the docstring shows the name of the type or method, in most cases how to call it and then gives some information explaining what it does and potentially hyperlinks to other similar docstrings ([`Optimizer`](@ref) in this case); all of this information is indented by a tab. Docstrings may include other information under subheaders **Arguments** (showing the arguments the method can be supplied with), **Examples** (giving more detailed examples (including results) of how to use the method) and **Implementation** (giving details on how the method is implemented). When we reference a docstring it is always printed in blue (e.g. [`optimization_step!`](@ref)), indicating a hyperlink. In addition there is an *index of docstrings* showing all docstrings in chronological order with the associated page number. Similar to **Library Functions**, which is included in most sections, almost every chapter concludes with a section **Chapter Summary** and an additional section **References** that shows further related reading material. The **Chapter Summary** recaps the important aspects of the corresponding chapter, states again what is new (this may be mathematical or software aspects) and gives information to what other parts of the dissertation the contents of the present chapter are relevant. @@ -35,11 +35,11 @@ This dissertation is structures into four main parts: (i) background information ## Background Information The background material, which does not include any original work, covers all the prerequisites for introducing our new optimizers in Part II. In addition it introduces some basic functionality of `GeometricMachineLearning`. It contains (among others) the following sections: -- [Concepts from general topology](@ref "Basic Concepts from General Topology"): here we introduce topological spaces, closedness, compactness, countability and Hausdorffness amongst others. These concepts are prerequisites for defining manifolds. -- [General theory on manifolds](@ref "(Matrix) Manifolds"): we introduce manifolds, the preimage theorem and submersion theorem. These theorems will be used to construct manifolds; the preimage theorem is used to give structure to the [Stiefel](@ref "The Stiefel Manifold") and the [Grassmann manifold](@ref "The Grassmann Manifold"), and the immersion theorem gives structure to the [solution manifold](@ref "The Solution Manifold") which is used in reduced order modeling. -- [Riemannian manifolds](@ref "Riemannian Manifolds"): for optimizing on manifolds we need to define a metric on them, which leads to *Riemannian manifolds*. We introduce *geodesics* and the *Riemannian gradient* here. -- [Homogeneous spaces](@ref "Homogeneous Spaces"): homogeneous spaces are a special class of manifolds to which our *generalized optimizer framework* can be applied. They trivially include all Lie groups and spaces like the [Stiefel manifold](@ref "The Stiefel Manifold"), the [Grassmann manifold](@ref "The Grassmann Manifold") and the "homogeneous space of positions and orientations" [bon2024optimal](@cite). -- [Global tangent spaces](@ref "Global Tangent Spaces"): homogeneous spaces allow for identifying for an invariant representation of all tangent spaces which we call *global tangent spaces*[^3]. We explain this concept in this section. +- [Concepts from general topology](@extref GeometricOptimizers Basic-Concepts-from-General-Topology): here we introduce topological spaces, closedness, compactness, countability and Hausdorffness amongst others. These concepts are prerequisites for defining manifolds. +- [General theory on manifolds](@extref GeometricOptimizers (Matrix)-Manifolds): we introduce manifolds, the preimage theorem and submersion theorem. These theorems will be used to construct manifolds; the preimage theorem is used to give structure to the [Stiefel](@extref GeometricOptimizers The-Stiefel-Manifold) and the [Grassmann manifold](@extref GeometricOptimizers The-Grassmann-Manifold), and the immersion theorem gives structure to the [solution manifold](@ref "The Solution Manifold") which is used in reduced order modeling. +- [Riemannian manifolds](@extref GeometricOptimizers Riemannian-Manifolds): for optimizing on manifolds we need to define a metric on them, which leads to *Riemannian manifolds*. We introduce *geodesics* and the *Riemannian gradient* here. +- [Homogeneous spaces](@extref GeometricOptimizers Homogeneous-Spaces): homogeneous spaces are a special class of manifolds to which our *generalized optimizer framework* can be applied. They trivially include all Lie groups and spaces like the [Stiefel manifold](@extref GeometricOptimizers The-Stiefel-Manifold), the [Grassmann manifold](@extref GeometricOptimizers The-Grassmann-Manifold) and the "homogeneous space of positions and orientations" [bon2024optimal](@cite). +- [Global tangent spaces](@extref GeometricOptimizers Global-Tangent-Spaces): homogeneous spaces allow for identifying for an invariant representation of all tangent spaces which we call *global tangent spaces*[^3]. We explain this concept in this section. - [Geometric structure](@ref "Symplectic Systems"): structure preservation takes a prominent role in this dissertation. In general *structure* refers to some property that the analytic solution of a differential equation also has and that we want to preserve when modeling the system. Here we discuss *symplecticity* and [volume preservation](@ref "Divergence-Free Vector Fields") in detail. We also introduce neural networks in this chapter and give a definition of [geometric neural networks](@ref "Structure-Preserving Neural Networks"). - [Reduced order modeling](@ref "Basic Concepts of Reduced Order Modeling"): reduced order modeling serves as a motivation for most of the architectures introduced here. In this section we introduce the basic idea behind reduced order modeling, show a typical workflow and explain what structure preservation looks like [in this context](@ref "Hamiltonian Model Order Reduction"). @@ -48,10 +48,10 @@ The background material, which does not include any original work, covers all th ## The Optimizer Framework One of the central parts of this dissertation is an *optimizer framework* that allows the generalization of existing optimizers such as Adam [kingma2014adam](@cite) to homogeneous spaces in a consistent way[^4]. This part contains the following sections: -- [Neural Network Optimizers](@ref): here we introduce the concept of a neural network optimizer and discuss the modifications we have to make in order to generalize them to homogeneous spaces. -- [Retractions](@ref): an important concept in manifold optimization are retractions [absil2008optimization](@cite). We introduce them in this section, discuss how they can be constructed for homogeneous spaces and show the two examples of the *geodesic retraction* and the *Cayley retraction*. -- [Parallel Transport](@ref): whenever we have an optimizer that contains momentum terms (such as Adam for example) we need to *transport* these momenta. In this section we explain how this can be done straightforwardly when dealing with homogeneous spaces. -- [Optimizer methods](@ref "Standard Neural Network Optimizers"): in this section we introduce simple optimizers such as the *gradient optimizer*, the *momentum optimizer* and *Adam* and show how to generalize them to our setting. +- [Neural network optimizers](@extref GeometricOptimizers The-optimizer-framework,-step-by-step): here we introduce the concept of a neural network optimizer and discuss the modifications we have to make in order to generalize them to homogeneous spaces. +- [Retractions](@extref GeometricOptimizers Retractions): an important concept in manifold optimization are retractions [absil2008optimization](@cite). We introduce them in this section, discuss how they can be constructed for homogeneous spaces and show the two examples of the *geodesic retraction* and the *Cayley retraction*. +- [Parallel transport](@extref GeometricOptimizers Parallel-Transport): whenever we have an optimizer that contains momentum terms (such as Adam for example) we need to *transport* these momenta. In this section we explain how this can be done straightforwardly when dealing with homogeneous spaces. +- [Optimizer methods](@extref GeometricOptimizers Standard-Neural-Network-Optimizers): in this section we introduce simple optimizers such as the *gradient optimizer*, the *momentum optimizer* and *Adam* and show how to generalize them to our setting. [^4]: The optimizer framework was introduced in [brantner2023generalizing](@cite). diff --git a/docs/src/layers/attention_layer.md b/docs/src/layers/attention_layer.md index 358c993e2..5f6a3569d 100644 --- a/docs/src/layers/attention_layer.md +++ b/docs/src/layers/attention_layer.md @@ -91,13 +91,13 @@ Besides the traditional attention mechanism `GeometricMachineLearning` therefore The Cayley transform maps from skew-symmetric matrices to orthonormal matrices. It takes the form[^4]: -[^4]: The Cayley transform here does not have the factor ``1/2`` hat we used when talking about the [Cayley retraction](@ref "Classical Retractions"). This is because now we do not need the retraction property ``d/dt\mathrm{Cayley}(tV)|_{t=0} = V``, but only a map ``\mathfrak{g}\to{}G=SO(N).`` +[^4]: The Cayley transform here does not have the factor ``1/2`` hat we used when talking about the [Cayley retraction](@extref GeometricOptimizers Classical-Retractions). This is because now we do not need the retraction property ``d/dt\mathrm{Cayley}(tV)|_{t=0} = V``, but only a map ``\mathfrak{g}\to{}G=SO(N).`` ```math \mathrm{Cayley}: A \mapsto (\mathbb{I} - A)(\mathbb{I} + A)^{-1}. ``` -Analogously to when we used the Cayley transform [as a retraction](@ref "Classical Retractions"), we can easily check that ``\mathrm{Cayley}(A)`` is orthogonal if ``A`` is skew-symmetric. For this consider ``\varepsilon \mapsto A(\varepsilon)\in\mathcal{S}_\mathrm{skew}`` with ``A(0) = \mathbb{O}`` and ``A'(0) = B \neq \mathbb{O}``. Then we have: +Analogously to when we used the Cayley transform [as a retraction](@extref GeometricOptimizers Classical-Retractions), we can easily check that ``\mathrm{Cayley}(A)`` is orthogonal if ``A`` is skew-symmetric. For this consider ``\varepsilon \mapsto A(\varepsilon)\in\mathcal{S}_\mathrm{skew}`` with ``A(0) = \mathbb{O}`` and ``A'(0) = B \neq \mathbb{O}``. Then we have: ```math \frac{\delta(\mathrm{Cayley}(A)^T\mathrm{Cayley}(A))}{\delta{}A} = \frac{d}{d\varepsilon}|_{\varepsilon=0} \mathrm{Cayley}(A(\varepsilon))^T \mathrm{Cayley}(A(\varepsilon)) = A'(0)^T + A'(0) = \mathbb{O}, @@ -112,7 +112,7 @@ For this the attention layer is modified in the following way: ```math Z := [z^{(1)}, \ldots, z^{(T)}] \mapsto Z\sigma(Z^TAZ), ``` -where ``\sigma(C)=\mathrm{Cayley}(C)`` and ``A`` is a matrix of type [`SkewSymMatrix`](@ref) that is learnable, i.e. the parameters of the attention layer are stored in ``A``. +where ``\sigma(C)=\mathrm{Cayley}(C)`` and ``A`` is a matrix of type [`SkewSymMatrix`](@extref GeometricOptimizers GeometricOptimizers.SkewSymMatrix) that is learnable, i.e. the parameters of the attention layer are stored in ``A``. ### Second approach: scalar products with an arbitrary weighting diff --git a/docs/src/layers/symplectic_attention.md b/docs/src/layers/symplectic_attention.md index 1c8cd0b9d..9cd96e145 100644 --- a/docs/src/layers/symplectic_attention.md +++ b/docs/src/layers/symplectic_attention.md @@ -32,7 +32,7 @@ Its gradient (with respect to ``Z``) is: \frac{\partial\Sigma(Z)}{\partial{}Z_{ij}} & = \frac{1}{1 + \sum{m, n}\exp(C_{mn})}\sum_{m'n'}\exp(C_{m'n'})\sum_{\ell}(\delta_{jm'}A_{i\ell}Z_{\ell{}n'} + \delta_{jn'}Z_{\ell{}m'}A_{\ell{}i}) \\ & = \frac{1}{1 + \sum_{m,n}\exp(C_{mn})}\{[AZ\exp.(C)^T]_{ij} + [A^TZ\exp.(C)]_{ij}\}. ``` -Note that if `A` is a [`SymmetricMatrix`](@ref) the expression than simplifies to: +Note that if `A` is a [`SymmetricMatrix`](@extref GeometricOptimizers GeometricOptimizers.SymmetricMatrix) the expression than simplifies to: ```math \frac{\partial\Sigma(Z)}{\partial{}Z_{ij}} = 2\frac{1}{1 + \sum_{m,n}\exp(C_{mn})}[AZ\exp.(C)^T]_{ij}, @@ -44,7 +44,7 @@ or written in matrix notation: \nabla_Z\Sigma(Z) = 2\frac{1}{1 + \sum_{m,n}\exp(C_{mn})}AZ\exp.(C). ``` -Whether we use a [`SymmetricMatrix`](@ref) for ``A`` or not can be set with the keyword `symmetric` in [`SymplecticAttention`](@ref). +Whether we use a [`SymmetricMatrix`](@extref GeometricOptimizers GeometricOptimizers.SymmetricMatrix) for ``A`` or not can be set with the keyword `symmetric` in [`SymplecticAttention`](@ref). ## Vector Softmax @@ -72,7 +72,7 @@ The first term is equivalent to: \mathrm{TermI:}\qquad \sum_n [AZ]_{in}[\mathrm{softmax}_1(C)^T]_{nj} \equiv AZ(\mathrm{softmax}_1(C))^T. ``` -If we again assume that the matrix `A` is a [`SymmetricMatrix`](@ref) then the expression simplifies to: +If we again assume that the matrix `A` is a [`SymmetricMatrix`](@extref GeometricOptimizers GeometricOptimizers.SymmetricMatrix) then the expression simplifies to: ```math \nabla_Z\Sigma(Z) = AZ\mathrm{softmax}_1(C). diff --git a/docs/src/layers/sympnet_gradient.md b/docs/src/layers/sympnet_gradient.md index 2da766992..d49e6b316 100644 --- a/docs/src/layers/sympnet_gradient.md +++ b/docs/src/layers/sympnet_gradient.md @@ -75,7 +75,7 @@ Linear layers of type ``p`` are of the form: \begin{pmatrix} q \\ p \end{pmatrix} \mapsto \begin{pmatrix} \mathbb{I} & \mathbb{O} \\ A & \mathbb{I} \end{pmatrix} \begin{pmatrix} q \\ p \end{pmatrix}, ``` -where ``A`` is a symmetric matrix. This is implemented very efficiently in `GeometricMachineLearning` with the special matrix [`SymmetricMatrix`](@ref). +where ``A`` is a symmetric matrix. This is implemented very efficiently in `GeometricMachineLearning` with the special matrix [`SymmetricMatrix`](@extref GeometricOptimizers GeometricOptimizers.SymmetricMatrix). ## Library Functions diff --git a/docs/src/layers/volume_preserving_feedforward.md b/docs/src/layers/volume_preserving_feedforward.md index 6f943370b..d85cac9a8 100644 --- a/docs/src/layers/volume_preserving_feedforward.md +++ b/docs/src/layers/volume_preserving_feedforward.md @@ -7,7 +7,7 @@ The *volume-preserving feedforward layers* in `GeometricMachineLearning` are clo ```math \mathtt{VPFF}_{A, b}: x \mapsto x + \sigma(Ax + b), ``` -where ``\sigma`` is a nonlinearity, ``A`` is the weight and ``b`` is the bias. The matrix ``A`` is either a [`LowerTriangular`](@ref) matrix ``L`` or an [`UpperTriangular`](@ref) matrix ``U``. We demonstrate volume-preservation of these layers by considering the case ``A = L``. The matrix looks as follows: +where ``\sigma`` is a nonlinearity, ``A`` is the weight and ``b`` is the bias. The matrix ``A`` is either a [`LowerTriangular`](@extref GeometricOptimizers GeometricOptimizers.LowerTriangular) matrix ``L`` or an [`UpperTriangular`](@extref GeometricOptimizers GeometricOptimizers.UpperTriangular) matrix ``U``. We demonstrate volume-preservation of these layers by considering the case ``A = L``. The matrix looks as follows: ```math L = \begin{pmatrix} @@ -28,7 +28,7 @@ J = \nabla\mathtt{VPFF}_{L, b} = \begin{pmatrix} b_{n1} & \cdots & b_{n(n-1)} & 1 \end{pmatrix}, ``` -and the determinant of ``J`` is 1, i.e. the map is volume-preserving. A similar statement holds if the matrix ``A`` is [`UpperTriangular`](@ref) instead of [`LowerTriangular`](@ref). +and the determinant of ``J`` is 1, i.e. the map is volume-preserving. A similar statement holds if the matrix ``A`` is [`UpperTriangular`](@extref GeometricOptimizers GeometricOptimizers.UpperTriangular) instead of [`LowerTriangular`](@extref GeometricOptimizers GeometricOptimizers.LowerTriangular). ## Library Functions diff --git a/docs/src/manifolds/basic_topology.md b/docs/src/manifolds/basic_topology.md deleted file mode 100644 index 6e99d4a2c..000000000 --- a/docs/src/manifolds/basic_topology.md +++ /dev/null @@ -1,159 +0,0 @@ -```@raw latex -% This is a summary of the manifold chapter; this is only visible in the latex version -In this chapter we introduce basic concepts necessary to discuss manifolds and manifold optimization. We begin by discussing \textit{topological vector spaces} and \textit{topological metric spaces}, and several theorems important for developing a theory of manifolds such as the \textit{implicit function theorem}. We then define manifolds and discuss the \textit{preimage theorem} and the \textit{immersion theorem} as tools to give general spaces the structure of a manifold. We then proceed with a discussion on \textit{geodesics} and \textit{Riemannian manifolds}. The chapter concludes with a presentation of \textit{homogeneous spaces} and their \textit{global tangent space representation} that will be crucial for generalizing neural network optimizers to the manifold setting. -``` - -# Basic Concepts from General Topology - -Here we discuss basic notions of topology that are necessary to define [manifolds](@ref "(Matrix) Manifolds") and work with them. Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a detailed discussion of the theory and for a wide range of examples that illustrate this theory see e.g. [lipschutz1965general](@cite). The here-presented concepts are also (rudimentarily) covered in most differential geometry textbooks such as [lang2012fundamentals, bishop1980tensor](@cite). - - -[^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes we require them here. - -We now start by giving all the definitions, theorem and corresponding proofs that are needed to define manifolds. Every manifold is a *topological space* which is why we give this definition first: - -```@eval -Main.definition(raw"A **topological space** is a set ``\mathcal{M}`` for which we are given a collection of subsets of ``\mathcal{M}``, which we denote by ``\mathcal{T}`` and call the *open subsets*. ``\mathcal{T}`` further has to satisfy the following three conditions: -" * -Main.indentation * raw"1. The empty set and ``\mathcal{M}`` belong to ``\mathcal{T}``. -" * -Main.indentation * raw"2. Any union of an arbitrary number of elements of ``\mathcal{T}`` again belongs to ``\mathcal{T}``. -" * -Main.indentation * raw"3. Any intersection of a finite number of elements of ``\mathcal{T}`` again belongs to ``\mathcal{T}``. -" * -Main.indentation * "So an arbitrary union of open sets is again open and a finite intersection of open sets is again open.") -``` - -Based on this definition of a topological space we can now define what it means to be *Hausdorff*: - -```@eval -Main.definition(raw"A topological space ``\mathcal{M}`` is said to be **Hausdorff** if for any two points ``x,y\in\mathcal{M}`` we can find two open sets ``U_x,U_y\in\mathcal{T}`` s.t. ``x\in{}U_x, y\in{}U_y`` and ``U_x\cap{}U_y=\{\}``.") -``` - -We now give the second definition that we need for defining manifolds, that of *second countability*: - -```@eval -Main.definition(raw"A topological space ``\mathcal{M}`` is said to be **second-countable** if we can find a countable subcollection of ``\mathcal{T}`` called ``\mathcal{U}`` s.t. ``\forall{}U\in\mathcal{T}`` and ``x\in{}U`` we can find an element ``V\in\mathcal{U}`` for which ``x\in{}V\subset{}U``.") -``` - -We now give a few definitions and results that are needed for the [inverse function theorem](@ref "The Inverse Function Theorem") which is essential for practical applications of manifold theory. We start with the definition of *continuity*: - -```@eval -Main.definition(raw"A mapping ``f`` between topological spaces ``\mathcal{M}`` and ``\mathcal{N}`` is called **continuous** if the preimage of every open set is again an open set, i.e. if ``f^{-1}\{U\}\in\mathcal{T}`` for ``U`` open in ``\mathcal{N}`` and ``\mathcal{T}`` the topology on ``\mathcal{M}``.") -``` - -Continuity can also be formulated in terms of *closed sets* instead of doing it with *open sets*. The definition of closed sets is given below: - -```@eval -Main.definition(raw"A **closed set** of a topological space ``\mathcal{M}`` is one whose complement is an open set, i.e. ``F`` is closed if ``F^c\in\mathcal{T}``, where the superscript ``{}^c`` indicates the complement: ``F^c := \{x\in\mathcal{M}:x\not\in{}F\}.`` For closed sets we thus have the following three properties: -" * -Main.indentation * raw"1. The empty set and ``\mathcal{M}`` are closed sets. -" * -Main.indentation * raw"2. Any union of a finite number of closed sets is again closed. -" * -Main.indentation * raw"3. Any intersection of an arbitrary number of closed sets is again closed. -" * -Main.indentation * "So a finite union of closed sets is again closed and an arbitrary intersection of closed sets is again closed.") -``` - -We now give the definition of continuity in terms of closed sets: - -```@eval -Main.theorem(raw"The definition of continuity in terms of open sets is equivalent to the following, second definition: ``f:\mathcal{M}\to\mathcal{N}`` is continuous if ``f^{-1}\{F\}\subset\mathcal{M}`` is a closed set for each closed set ``F\subset\mathcal{N}``.") -``` - -```@eval -Main.proof(raw"First assume that ``f`` is continuous according to the first definition and not to the second. Then ``f^{-1}\{F\}`` is not closed but ``f^{-1}\{F^c\}`` is open. But ``f^{-1}\{F^c\} = \{x\in\mathcal{M}:f(x)\not\in\mathcal{N}\} = (f^{-1}\{F\})^c`` cannot be open, else ``f^{-1}\{F\}`` would be closed. The implication of the first definition under assumption of the second can be shown analogously.") -``` - -The next theorem makes the rather abstract definition of *closed sets* more concrete; this definition is especially important for many practical proofs: - -```@eval -Main.theorem(raw"The property of a set ``F`` being closed is equivalent to the following statement: If a point ``y`` is such that for every open set ``U`` containing it we have ``U\cap{}F\ne\{\}`` then this point is contained in ``F``.") -``` - -```@eval -Main.proof(raw"We first proof that if a set is closed then the statement holds. Consider a closed set ``F`` and a point ``y\not\in{}F`` s.t. every open set containing ``y`` has nonempty intersection with ``F``. But the complement ``F^c`` also is such a set, which is a clear contradiction. Now assume the above statement for a set ``F`` and further assume ``F`` is not closed. Its complement ``F^c`` is thus not open. Now consider the *interior* of this set: ``\mathrm{int}(F^c):=\cup\{U:U\subset{}F^c\text{ and $U$ open}\}``, i.e. the biggest open set contained within ``F^c``. Hence there must be a point ``y`` which is in ``F^c`` but is not in its interior, else ``F^c`` would be equal to its interior, i.e. would be open. We further must be able to find an open set ``U`` that contains ``y`` but is also contained in ``F^c``, else ``y`` would be an element of ``F``. A contradiction.") -``` - -Next we define *open covers*, a concept that is very important in developing a theory of manifolds: - -```@eval -Main.definition(raw"An **open cover** of a topological space ``\mathcal{M}`` is a (not necessarily countable) collection of open sets ``\{U_i\}_{i\mathcal{I}}`` s.t. their union contains ``\mathcal{M}``. A **finite open cover** is a finite collection of open sets that cover ``\mathcal{M}``. We say that an open cover is **reducible** to a finite cover if we can find a finite number of elements in the open cover whose union still contains ``\mathcal{M}``.") -``` - -And connected to this definition we state what it means for a topological space to be *compact*. This is a rather strong property that some of the manifolds treated in here have, for example the [Stiefel manifold](@ref "The Stiefel Manifold"). - -```@eval -Main.definition(raw"A topological space ``\mathcal{M}`` is called **compact** if every open cover is reducible to a finite cover.") -``` - -A very important result from general topology is that continuous functions preserve compactness[^2]: - -[^2]: We also say that *compactness is a topological property* [lipschutz1965general](@cite). - -```@eval -Main.theorem(raw"Consider a continuous function ``f:\mathcal{M}\to\mathcal{N}`` and a compact set ``K\in\mathcal{M}``. Then ``f(K)`` is also compact.") -``` - -```@eval -Main.proof(raw"Consider an open cover of ``f(K)``: ``\{U_i\}_{i\in\mathcal{I}}``. Then ``\{f^{-1}\{U_i\}\}_{i\in\mathcal{I}}`` is an open cover of ``K`` and hence reducible to a finite cover ``\{f^{-1}\{U_i\}\}_{i\in\{i_1,\ldots,i_n\}}``. But then ``\{{U_i\}_{i\in\{i_1,\ldots,i_n}}`` also covers ``f(K)``.") -``` - -Moreover compactness is a property that is *inherited* by closed subspaces: - -```@eval -Main.theorem(raw"A closed subset of a compact space is compact.") -``` - -```@eval -Main.proof(raw"Call the closed set ``F`` and consider an open cover of this set: ``\{U\}_{i\in\mathcal{I}}``. Then this open cover combined with ``F^c`` is an open cover for the entire compact space, hence reducible to a finite cover.") -``` - -If a set is contained in a Hausdorff space and is also compact we have: - -```@eval -Main.theorem(raw"A compact subset of a Hausdorff space is closed.") -``` - -```@eval -Main.proof(raw"Consider a compact subset ``K``. If ``K`` is not closed, then there has to be a point ``y\not\in{}K`` s.t. every open set containing ``y`` intersects ``K``. Because the surrounding space is Hausdorff we can now find the following two collections of open sets: ``\{(U_z, U_{z,y}: U_z\cap{}U_{z,y}=\{\})\}_{z\in{}K}``. The open cover ``\{U_z\}_{z\in{}K}`` is then reducible to a finite cover ``\{U_z\}_{z\in\{z_1, \ldots, z_n\}}``. The intersection ``\cap_{z\in{z_1, \ldots, z_n}}U_{z,y}`` is then an open set that contains ``y`` but has no intersection with ``K``. A contraction.") -``` - -This last theorem we will use in proofing the [inverse function theorem](@ref "The Inverse Function Theorem"): - -```@eval -Main.theorem(raw"If ``\mathcal{M}`` is compact and ``\mathcal{N}`` is Hausdorff, then the inverse of a continuous injective function ``f:\mathcal{M}\to\mathcal{N}`` is again continuous, i.e. ``f(V)`` is an open set in ``\mathcal{N}`` for ``V\in\mathcal{T}``.") -``` - -```@eval -Main.proof(raw"We can equivalently show that every closed set is mapped to a closed set. First consider the set ``K\in\mathcal{M}``. Its image is again compact and hence closed because ``\mathcal{N}`` is Hausdorff.") -``` - -We further define what it means for a set to be *dense*: - -```@eval -Main.definition(raw"A set ``U`` is called **dense in ``D``**, where ``U\subset{}D`` if the *closure of ``U``*, i.e. the smallest closed set containing ``U``, also contains ``D``.") -``` - -We will come back to the notion of *denseness* when talking about the [universal approximation theorem for SympNets](@ref "Universal Approximation Theorems"). - -```@raw latex -\begin{comment} -``` - -## References - -references = raw""" -```@bibliography -Pages = [] -Canonical = false - -lipschutz1965general -lang2012fundamentals -bishop1980tensor -``` - -```@raw latex -\end{comment} -``` \ No newline at end of file diff --git a/docs/src/manifolds/existence_and_uniqueness_theorem.md b/docs/src/manifolds/existence_and_uniqueness_theorem.md deleted file mode 100644 index 4967d7fca..000000000 --- a/docs/src/manifolds/existence_and_uniqueness_theorem.md +++ /dev/null @@ -1,81 +0,0 @@ -# The Existence-And-Uniqueness Theorem - -The *existence-and-uniqueness theorem*, also known as the *Picard-Lindelöf theorem*, *Picard's existence theorem* or the *Cauchy-Lipschitz theorem* gives a proof of the existence of solutions for ODEs. Here we state the existence-and-uniqueness theorem for manifolds as vector spaces are just a special case of this. Its proof relies on the [Banach fixed-point theorem](@ref "The Fixed-Point Theorem")[^1]. - -[^1]: It has to be noted that the proof given here is not entirely self-contained. The proof of the fundamental theorem of calculus, i.e. the proof of the existence of an antiderivative of a continuous function [lang2012real](@cite), is omitted for example. - -```@eval -Main.theorem(raw"Let ``X`` a vector field on the manifold ``\mathcal{M}`` that is differentiable at ``x``. Then we can find an ``\epsilon>0`` and a unique curve ``\gamma:(-\epsilon, \epsilon)\to\mathcal{M}`` such that ``\gamma'(t) = X(\gamma(t))``."; name = "Existence-And-Uniqueness Theorem") -``` - -```@eval -Main.proof(raw"We consider a ball around a point ``x\in\mathcal{M}`` with radius ``r`` that we pick such that the ball ``B(x, r)`` fits into the ``U`` of some coordinate chart ``\varphi_U``; we further use ``X`` and ``\varphi'\circ{}X\circ\varphi^{-1}`` interchangeably in this proof. We then define ``L := \mathrm{sup}_{y,z\in{}B(x,r)}|X(y) - X(z)|/|y - z|.`` Note that this ``L`` is always finite because ``X`` is bounded and differentiable. We now define the map ``\Gamma: C^\infty((-\epsilon, \epsilon), \mathbb{R}^n)\to{}C^\infty((-\epsilon, \epsilon), \mathbb{R}^n)`` (for some ``\epsilon`` that we do not yet fix) as -" * -Main.indentation * raw"```math -" * -Main.indentation * raw"\Gamma\gamma(t) = x + \int_0^tX(\gamma(s))ds, -" * -Main.indentation * raw"``` -" * -Main.indentation * raw"i.e. ``\Gamma`` maps ``C^\infty`` curves through ``x`` into ``C^\infty`` curves through ``x``. We further have with the norm ``||\gamma||_\infty = \mathrm{sup}_{t \in (-\epsilon, \epsilon)}|\gamma(t)|``: -" * -Main.indentation * raw"```math -" * -Main.indentation * raw"\begin{aligned} -" * -Main.indentation * raw"||\Gamma(\gamma_1 - \gamma_2)||_\infty & = \mathrm{sup}_{t \in (-\epsilon, \epsilon)}\left| \int_0^t (X(\gamma_1(s)) - X(\gamma_2(s)))ds \right| \\ -" * -Main.indentation * raw"& \leq \mathrm{sup}_{t \in (-\epsilon, \epsilon)}\int_0^t | X(\gamma_1(s)) - X(\gamma_2(s)) | ds \\ -" * -Main.indentation * raw"& \leq \mathrm{sup}_{t \in (-\epsilon, \epsilon)}\int_0^t L |\gamma_1(s) - \gamma_2(s)| ds \\ -" * -Main.indentation * raw"& \leq \epsilon{}L \cdot \mathrm{sup}_{t \in (-\epsilon, \epsilon)}|\gamma_1(t) - \gamma_2(t)|, -" * -Main.indentation * raw"\end{aligned} -" * -Main.indentation * raw"``` -" * -Main.indentation * raw"and we see that ``\Gamma`` is a contractive mapping if we pick ``\epsilon`` small enough and we can hence apply the fixed-point theorem. So there has to exist a ``C^\infty`` curve through ``x`` that we call ``\gamma^*`` such that -" * -Main.indentation * raw"```math -" * -Main.indentation * raw"\gamma^*(t) = \int_0^tX(\gamma^*(s))ds, -" * Main.indentation * raw"``` -" * Main.indentation * raw"and this ``\gamma^*`` is the curve we were looking for. Its uniqueness is guaranteed by the fixed-point theorem.") -``` - -For all the problems we discuss here we can extend the integral curves of ``X`` from the finite interval ``(-\epsilon, \epsilon)`` to all of ``\mathbb{R}``. The solution ``\gamma`` we call an *integral curve* or *flow* of the vector field (ODE). - -## Time-Dependent Vector Fields - -We proved the theorem above for a time-independent vector field ``X``, but it also holds for time-dependent vector fields, i.e. for mappings of the form: - -```math -X: [0,T]\times\mathcal{M}\to{}TM. -``` - -The proof for this case proceeds analogously to the case of the time-independent vector field; to apply the proof we simply have to *extend* the vector field to (here written for a specific coordinate chart ``\varphi_U``): - -```math -\bar{X}: [0, T]\times\mathbb{R}^n\to{}\mathbb{R}^{n+1},\, (t, x_1, \ldots, x_n) \mapsto (1, X(x_1, \ldots, x_n)). -``` - -More details on this can be found in e.g. [lang2012fundamentals](@cite). For `GeometricMachineLearning` time-dependent vector fields are important because many of the optimizers we are using (such as the [Adam optimizer](@ref "The Adam Optimizer")) can be seen as approximating the flow of a time-dependent vector field. - -```@raw latex -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -lang2012real -lang2012fundamentals -``` - -```@raw latex -\end{comment} -``` \ No newline at end of file diff --git a/docs/src/manifolds/homogeneous_spaces.md b/docs/src/manifolds/homogeneous_spaces.md deleted file mode 100644 index 39215894d..000000000 --- a/docs/src/manifolds/homogeneous_spaces.md +++ /dev/null @@ -1,244 +0,0 @@ -# Homogeneous Spaces - -*Homogeneous spaces* are very important in `GeometricMachineLearning` as we can generalize existing neural network optimizers from vector spaces to such homogenous spaces. They are intricately linked to the notion of a *Lie Group* and its *Lie Algebra*[^1]. - -[^1]: Recall that a Lie group is a manifold that also has group structure. We say that a Lie group ``G`` *acts* on a manifold ``\mathcal{M}`` if there is a map ``G\times\mathcal{M} \to \mathcal{M}`` such that ``(ab)x = a(bx)`` for ``a,b\in{}G`` and ``x\in\mathcal{M}``. For us the Lie algebra belonging to a Lie group, denoted by ``\mathfrak{g}``, is the tangent space to the identity element ``T_\mathbb{I}G``. - -```@eval -Main.definition(raw"A **homogeneous space** is a manifold ``\mathcal{M}`` on which a Lie group ``G`` acts transitively, i.e. -" * Main.indentation * raw" ```math -" * Main.indentation * raw"\forall X,Y\in\mathcal{M} \quad \exists{}A\in{}G\text{ s.t. }AX = Y. -" * Main.indentation * raw"``` -") -``` - -Now fix a distinct element ``E\in\mathcal{M}``; we will refer to this as the *canonical element* or [`StiefelProjection`](@ref). We can also establish an isomorphism between ``\mathcal{M}`` and the quotient space ``G/\sim`` with the equivalence relation: -```math -A_1 \sim A_2 \iff A_1E = A_2E. -``` -Note that this is independent of the chosen ``E``. - -The tangent spaces of ``\mathcal{M}`` are of the form ``T_Y\mathcal{M} = \mathfrak{g}\cdot{}Y``, i.e. can be fully described through its Lie algebra. -Based on this we can perform a splitting of ``\mathfrak{g}`` into two parts: - -```@eval -Main.definition(raw"A **splitting of the Lie algebra** ``\mathfrak{g}`` at an element of a homogeneous space ``Y`` is a decomposition into a **vertical** and a **horizontal** component, denoted by ``\mathfrak{g} = \mathfrak{g}^{\mathrm{ver},Y} \oplus \mathfrak{g}^{\mathrm{hor},Y}`` such that -" * Main.indentation * raw"1. The **vertical component** ``\mathfrak{g}^{\mathrm{ver},Y}`` is the kernel of the map ``\mathfrak{g}\to{}T_Y\mathcal{M}, V \mapsto VY``, i.e. ``\mathfrak{g}^{\mathrm{ver},Y} = \{V\in\mathfrak{g}:VY = 0\}.`` -" * Main.indentation * raw"2. The **horizontal component** ``\mathfrak{g}^{\mathrm{hor},Y}`` is the orthogonal complement of ``\mathfrak{g}^{\mathrm{ver},Y}`` in ``\mathfrak{g}``. It is isomorphic to ``T_Y\mathcal{M}``. -" * Main.indentation * raw"*Orthogonal complement* means that ``\forall{}V\in\mathfrak{g}^{\mathrm{ver}, Y}`` and ``\forall{}B\in\mathfrak{g}^{\mathrm{hor}, Y}`` we have ``\langle V, B \rangle = 0`` for some metric ``\langle\cdot,\cdot\rangle`` defined on ``\mathfrak{g}``.") -``` - -We will refer to the isomorphism from ``T_Y\mathcal{M}`` to ``\mathfrak{g}^{\mathrm{hor}, Y}`` by ``\Omega``. We will give explicit examples of ``\Omega`` below. The metric ``\langle\cdot,\cdot\rangle`` on ``\mathfrak{g}`` further induces a Riemannian metric on ``\mathcal{M}``: -```math -g_Y(\Delta_1, \Delta_2) = \langle\Omega(Y,\Delta_1),\Omega(Y,\Delta_2)\rangle\text{ for $\Delta_1,\Delta_2\in{}T_Y\mathcal{M}$.} -``` - -Two examples of homogeneous spaces implemented in `GeometricMachineLearning` are the [Stiefel manifold](@ref "The Stiefel Manifold") and the [Grassmann manifold](@ref "The Grassmann Manifold"). The Lie group ``SO(N)`` acts transitively on both of these manifolds, i.e. turns them into homogeneous spaces. We give its Lie algebra as an example here: - -```@eval -Main.example(raw"The Lie algebra of ``SO(N)`` are the skew-symmetric matrices ``\mathfrak{so}(N):=\{V\in\mathbb{R}^{N\times{}N}:V^T + V = 0\}`` and the canonical metric associated with it is simply ``(V_1,V_2)\mapsto\frac{1}{2}\mathrm{Tr}(V_1^TV_2)``.") -``` - - -# The Stiefel Manifold - -The Stiefel manifold ``St(n, N)`` is the space of all orthonormal frames in ``\mathbb{R}^{N\times{}n}``, i.e. matrices ``Y\in\mathbb{R}^{N\times{}n}`` s.t. ``Y^TY = \mathbb{I}_n``. It can also be seen as ``SO(N)`` modulo an equivalence relation: ``A\sim{}B\iff{}AE = BE`` for - -```math -E = \begin{bmatrix} -\mathbb{I}_n \\ -\mathbb{O} -\end{bmatrix}\in{}St(n, N), -``` -which is the canonical element of the Stiefel manifold that we call [`StiefelProjection`](@ref). In words: the first ``n`` columns of ``A`` and ``B`` are the same. We also use this principle to draw random elements from the Stiefel manifold. - -```@eval -Main.remark(raw"Drawing random elements from the Stiefel (and the Grassmann) manifold is done by first calling `rand(N, n)` (i.e. drawing from a normal distribution) and then performing a ``QR`` decomposition. We then take the first ``n`` columns of the ``Q`` matrix to be an element of the Stiefel manifold.") -``` - -The tangent space to the element ``Y\in{}St(n,N)`` can be determined by considering ``C^\infty`` curves on ``SO(N)`` through ``\mathbb{I}.`` We write those curves as ``t\mapsto{}A(t)``. Because ``SO(N)`` acts transitively on ``St(n, N)`` each ``C^\infty`` curve on ``St(n, N)`` through ``Y`` can be written as ``A(t)Y`` and we get: - -```math -T_YSt(n,N)=\{BY : B\in\mathfrak{g}\} = \{\Delta\in\mathbb{R}^{N\times{}n}: \Delta^TY + Y^T\Delta = \mathbb{O}\}, -``` - -where the last equality[^2] can be established through the isomorphism: - -[^2]: Note that we can easily check ``\{BY : B\in\mathfrak{g}\} \subset \{\Delta\in\mathbb{R}^{N\times{}n}: \Delta^TY + Y^T\Delta = \mathbb{O}\}.`` The isomorphism is used to proof ``\{\Delta\in\mathbb{R}^{N\times{}n}: \Delta^TY + Y^T\Delta = \mathbb{O}\} \subset \{BY : B\in\mathfrak{g}\}`` as ``\Omega(\Delta)\in\mathfrak{g}^{\mathrm{hor},Y}\subset\mathfrak{g}`` and ``\Delta = \Omega(\Delta)Y.`` - -```math -\Omega: T_YSt(n, N) \to \mathfrak{g}^{\mathrm{hor}, Y}, \Delta \mapsto (\mathbb{I} - \frac{1}{2}YY^T)\Delta{}Y^T - Y\Delta^T(\mathbb{I} - \frac{1}{2}YY^T). -``` - -That this is an isomorphism can be easily checked: - -```math - \Omega(\Delta)Y = (\mathbb{I} - \frac{1}{2}YY^T)\Delta - \frac{1}{2}Y\Delta^TY = \Delta. -``` - -This isomorphism is implemented in `GeometricMachineLearning`: - -```@example omega_metric -using GeometricMachineLearning # hide -using GeometricMachineLearning: Ω -Y = rand(StiefelManifold, 5, 3) -Δ = rgrad(Y, rand(5, 3)) -@assert Ω(Y, Δ) * Y.A ≈ Δ # hide -Ω(Y, Δ) * Y.A ≈ Δ -``` - -The function [`rgrad`](@ref), which maps ``\mathbb{R}^{N\times{}n}`` to ``T_YSt(n, N)`` is introduced below. We can now also introduce the Riemannian metric on ``St(n,N)``: - -```math -g_Y(\Delta_1, \Delta_2) = \mathrm{Tr}\left( \frac{1}{2} \Omega(\Delta_1)^T \Omega(\Delta_2) \right) = \mathrm{Tr}(\Delta_1^T(\mathbb{I} - \frac{1}{2}YY^T)\Delta_2). -``` - -We can check that this is true: - -```@example omega_metric -using LinearAlgebra: tr -Δ₂ = rgrad(Y, rand(5, 3)) -@assert .5 * tr(Ω(Y, Δ)' * Ω(Y, Δ₂)) ≈ metric(Y, Δ, Δ₂) # hide -.5 * tr(Ω(Y, Δ)' * Ω(Y, Δ₂)) ≈ metric(Y, Δ, Δ₂) -``` - -## The Riemannian Gradient for the Stiefel Manifold - -We defined the [Riemannian gradient](@ref "The Riemannian Gradient") to be a vector field ``\mathrm{grad}^gL`` such that it is *compatible with the Riemannian metric* in some sense; the definition we gave relied on an explicit coordinate chart. We can also express the Riemannian gradient for matrix manifolds by not relying on an explicit coordinate representation (which would be computationally expensive) [absil2004riemannian](@cite). - -```@eval -Main.definition(raw"Given a Riemannian matrix manifold ``\mathcal{M}`` we define the **Riemannian gradient** of ``L:\mathcal{M}\to\mathbb{R}`` at ``Y``, called ``\mathrm{grad}_YL\in{}T_Y\mathcal{M}``, as the unique element of ``T_Y\mathcal{M}`` such that for any other ``\Delta\in{}T_Y\mathcal{M}`` we have -" * Main.indentation * raw"```math -" * Main.indentation * raw"\mathrm{Tr}((\nabla{}L)^T\Delta) = g_Y(\mathrm{grad}_YL, \Delta), -" * Main.indentation * raw"``` -" * Main.indentation * raw"where Tr indicates the usual matrix trace.") -``` - -For the Stiefel manifold the Riemannian gradient is given by: - -```math - \mathrm{grad}_YL = \nabla_YL - Y(\nabla_YL)^TY =: \mathtt{rgrad}(Y, \nabla_YL), -``` - -where ``\nabla_YL`` refers to the Euclidean gradient, i.e. - -```math - [\nabla_YL]_{ij} = \frac{\partial{}L}{\partial{}y_{ij}}. -``` - -The Euclidean gradient ``\nabla{}L`` can in practice be obtained with an [AD routine](@ref "Pullbacks and Automatic Differentiation"). We then use the function [`rgrad`](@ref) to map ``\nabla_YL`` from ``\mathbb{R}^{N\times{}n}`` to ``T_YSt(n,N)``. We can check that this mapping indeed produces the Riemannian gradient[^3]: - -[^3]: Here we are testing with a randomly drawn element ``\Delta\in{}T_Y\mathcal{M}.`` - -```@example -using GeometricMachineLearning # hide -using LinearAlgebra: tr - -Y = rand(StiefelManifold, 5, 3) -∇L = rand(5, 3) -gradL = rgrad(Y, ∇L) -Δ = rgrad(Y, rand(5, 3)) - -@assert metric(Y, gradL, Δ) ≈ tr(∇L' * Δ) # hide -metric(Y, gradL, Δ) ≈ tr(∇L' * Δ) -``` - -# The Grassmann Manifold - -The Grassmann manifold is closely related to the Stiefel manifold, and an element of the Grassmann manifold can be represented through an element of the Stiefel manifold (but not vice-versa). An element of the Grassmann manifold ``Gr(n,N)`` is a vector subspace ``\subset\mathbb{R}^N`` of dimension $n$. Each such subspace (i.e. element of the Grassmann manifold) can be represented by a full-rank matrix ``A\in\mathbb{R}^{N\times{}n}`` and we identify two elements with the following equivalence relation: - -```math - A_1 \sim A_2 \iff \exists{}C\in\mathbb{R}^{n\times{}n}\text{ s.t. }A_1C = A_2. -``` - -The resulting manifold is of dimension ``n(N-n)``. One can find a parametrization of the manifold the following way: Because the matrix ``Y`` has full rank, there have to be ``n`` independent rows in it: ``i_1, \ldots, i_n``. For simplicity assume that ``i_1 = 1, i_2=2, \ldots, i_n=n`` and call the matrix made up of these columns ``C``. Then the mapping to the coordinate chart is: ``YC^{-1}`` and the last ``N-n`` columns are the coordinates. - -We can also define the Grassmann manifold based on the Stiefel manifold since elements of the Stiefel manifold are already full-rank matrices. In this case we have the following equivalence relation (for ``Y_1, Y_2\in{}St(n,N)``): - -```math - Y_1 \sim Y_2 \iff \exists{}C\in{}SO(n)\text{ s.t. }Y_1C = Y_2. -``` - -In `GeometricMachineLearning` elements of the Grassmann manifold are drawn the same way as elements of the Stiefel manifold: - -```@example -using GeometricMachineLearning # hide -rand(GrassmannManifold{Float32}, 5, 3) -``` - -## The Riemannian Gradient of the Grassmann Manifold - -Obtaining the Riemannian Gradient for the Grassmann manifold is slightly more difficult than it is in the case of the Stiefel manifold [absil2004riemannian](@cite). Since the Grassmann manifold can be obtained from the Stiefel manifold through an equivalence relation, we can however use this as a starting point. - -```@eval -Main.theorem(raw"The Riemannian gradient of a function ``L`` defined on the Grassmann manifold can be written as -" * Main.indentation * raw"```math -" * Main.indentation * raw"\mathrm{grad}_\mathcal{Y}^{Gr}L \simeq \nabla_Y{}L - YY^T\nabla_YL, -" * Main.indentation * raw"``` -" * Main.indentation * raw"where ``\nabla_Y{}L`` is again the Euclidean gradient.") -``` - -```@eval -Main.proof(raw"In a first step we identify charts on the Grassmann manifold to make dealing with it easier. For this consider the following open cover of the Grassmann manifold. -" * Main.indentation * raw"```math -" * Main.indentation * raw"\{\mathcal{U}_W\}_{W\in{}St(n, N)} \quad\text{where}\quad \mathcal{U}_W = \{\mathrm{span}(Y):\mathrm{det}(W^TY)\neq0\}. -" * Main.indentation * raw"``` -" * Main.indentation * raw"We can find a canonical bijective mapping from the set ``\mathcal{U}_W`` to the set ``\mathcal{S}_W := \{Y\in\mathbb{R}^{N\times{}n}:W^TY=\mathbb{I}_n\}``: -" * Main.indentation * raw"```math -" * Main.indentation * raw"\sigma_W: \mathcal{U}_W \to \mathcal{S}_W,\, \mathcal{Y}=\mathrm{span}(Y)\mapsto{}Y(W^TY)^{-1} =: \hat{Y}. -" * Main.indentation * raw"``` -" * Main.indentation * raw"That ``\sigma_W`` is well-defined is easy to see: Consider ``YC`` with ``C\in\mathbb{R}^{n\times{}n}`` non-singular. Then ``YC(W^TYC)^{-1}=Y(W^TY)^{-1} = \hat{Y}``. With this isomorphism we can also find a representation of elements of the tangent space: -" * Main.indentation * raw"```math -" * Main.indentation * raw"T_\mathcal{Y}\sigma_W: T_\mathcal{Y}Gr(n,N)\to{}T_{\hat{Y}}\mathcal{S}_W. -" * Main.indentation * raw"``` -" * Main.indentation * raw"We give an explicit representation of this isomorphism; because the map ``\sigma_W`` does not care about the representation of ``\mathrm{span}(Y)`` we can perform the variations in ``St(n,N)``. We write the variations as ``Y(t)\in{}St(n,N)`` for ``t\in(-\varepsilon,\varepsilon)``. We also set ``Y(0) = Y`` and hence -" * Main.indentation * raw"```math -" * Main.indentation * raw"\frac{d}{dt}Y(t)(W^TY(t))^{-1} = (\dot{Y}(0) - Y(W^TY)^{-1}W^T\dot{Y}(0))(W^TY)^{-1}, -" * Main.indentation * raw"``` -" * Main.indentation * raw"where ``\dot{Y}(0)\in{}T_YSt(n,N)``. Also note note that we have ``T_\mathcal{Y}\mathcal{U}_W = T_\mathcal{Y}Gr(n,N)`` because ``\mathcal{U}_W`` is an open subset of ``Gr(n,N)``. We thus can identify the tangent space ``T_\mathcal{Y}Gr(n,N)`` with the following set: -" * Main.indentation * raw"```math -" * Main.indentation * raw"T_{\hat{Y}}\mathcal{S}_W = \{(\Delta - YW^T\Delta)(W^TY)^{-1}: Y\in{}St(n,N)\text{ s.t. }\mathrm{span}(Y)=\mathcal{Y}\text{ and }\Delta\in{}T_YSt(n,N)\}. -" * Main.indentation * raw"``` -" * Main.indentation * raw"Further note that we can pick any element ``W`` to construct the charts for a neighborhood around the point ``\mathcal{Y}\in{}Gr(n,N)`` as long as we have ``\mathrm{det}(W^TY)\neq0`` for ``\mathrm{span}(Y)=\mathcal{Y}``. We hence take ``W=Y`` and get the identification: -" * Main.indentation * raw"```math -" * Main.indentation * raw"T_\mathcal{Y}Gr(n,N) \equiv \{\Delta - YY^T\Delta: Y\in{}St(n,N)\text{ s.t. }\mathrm{span}(Y)=\mathcal{Y}\text{ and }\Delta\in{}T_YSt(n,N)\}, -" * Main.indentation * raw"``` -" * Main.indentation * raw"which is very easy to handle computationally (we simply store and change the matrix ``Y`` that represents an element of the Grassmann manifold). In this representation the Riemannian gradient is then -" * Main.indentation * raw"```math -" * Main.indentation * raw"\mathrm{grad}_\mathcal{Y}^{Gr}L = \mathrm{grad}_Y^{St}L - YY^T\mathrm{grad}_Y^{St}L = \nabla_Y{}L - YY^T\nabla_YL, -" * Main.indentation * raw"``` -" * Main.indentation * raw"where ``\mathrm{grad}^{St}_YL`` is the Riemannian gradient of the Stiefel manifold at ``Y``. We proved our assertion.") -``` - - ## Library Functions - -```@docs -StiefelManifold -StiefelProjection -GrassmannManifold -GeometricMachineLearning.metric(::StiefelManifold, ::AbstractMatrix, ::AbstractMatrix) -GeometricMachineLearning.rgrad(::StiefelManifold, ::AbstractMatrix) -GeometricMachineLearning.metric(::GrassmannManifold, ::AbstractMatrix, ::AbstractMatrix) -GeometricMachineLearning.rgrad(::GrassmannManifold, ::AbstractMatrix) -GeometricMachineLearning.Ω(::StiefelManifold{T}, ::AbstractMatrix{T}) where T -GeometricMachineLearning.Ω(::GrassmannManifold{T}, ::AbstractMatrix{T}) where T -``` - -```@raw latex -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -absil2004riemannian -frankel2011geometry -bendokat2021real -``` - -```@raw latex -\end{comment} -``` \ No newline at end of file diff --git a/docs/src/manifolds/inverse_function_theorem.md b/docs/src/manifolds/inverse_function_theorem.md deleted file mode 100644 index 90fd7a02d..000000000 --- a/docs/src/manifolds/inverse_function_theorem.md +++ /dev/null @@ -1,99 +0,0 @@ -# Foundational Theorems for Differential Manifolds - -Here we state and proof all the theorems necessary to define [differential manifolds](@ref "(Matrix) Manifolds"). All these theorems (including proofs) can be found in e.g. [lang2012fundamentals](@cite). - -## The Fixed-Point Theorem - -The fixed-point theorem will be used in the proof of the inverse function theorem below and the [existence-and-uniqueness theorem](@ref "The Existence-And-Uniqueness Theorem"). - -```@eval -Main.theorem(raw"A function ``f:U \to U`` defined on an open subset ``U`` of a complete metric vector space ``\mathcal{V} \supset U`` that is contractive, i.e. ``|f(z) - f(y)| \leq q|z - y|`` with ``q < 1``, has a unique fixed point ``y^*`` such that ``f(y^*) = y^*``. Further ``y^*`` can be found by taking any ``y\in{}U`` through ``y^* = \lim_{m\to\infty}f^m(y)``."; name = "Banach Fixed-Point Theorem") -``` - -```@eval -Main.proof(raw"Fix a point ``y\in{}U``. We proof that the sequence ``(f^m(y))_{m\in\mathbb{N}}`` is Cauchy and because ``\mathcal{V}`` is a complete metric space, the limit of this sequence exists. Take ``\tilde{m} > m`` and we have -" * -Main.indentation * raw"```math -" * -Main.indentation * raw"\begin{aligned} -" * -Main.indentation * raw"|f^{\tilde{m}}(y) - f^m(y)| & \leq \sum_{i = m}^{\tilde{m} - 1}|f^{i+1}(y) - f^{i}(y)| \\ -" * -Main.indentation * raw" & \leq \sum_{i = m}^{\tilde{m} - 1}q^i|f(y) - y| \\ -" * -Main.indentation * raw" & \leq \sum_{i = m}^\infty{}q^i|f(y) - y| = (f(y) - y)\left( \frac{q}{1 - q} - \sum_{i = 1}^{m-1}q^i \right)\\ -" * -Main.indentation * raw" & = (f(y) - y)\left( \frac{q}{1 - q} - \frac{q - q^m}{q - 1} \right) = (f(y) - y)\frac{q^{m+1}}{1 - q}. -" * -Main.indentation * raw"\end{aligned} -" * -Main.indentation * raw"``` -" * -Main.indentation * raw"And the sequence is clearly Cauchy.") -``` - -Note that we stated the fixed-point theorem for arbitrary complete metric spaces here, not just for ``\mathbb{R}^n``. For the section on [manifolds](@ref "(Matrix) Manifolds") we only need the theorem for ``\mathbb{R}^n``, but for the [existence-and-uniqueness theorem](@ref "The Existence-And-Uniqueness Theorem") we need the statement for more general spaces. - - -## The Inverse Function Theorem - -The *inverse function theorem* gives a sufficient condition on a vector-valued function to be invertible in a neighborhood of a specific point. This theorem serves as a basis for the *implicit function theorem* and further for the [preimage theorem](@ref "The Preimage Theorem") and is critical in developing a theory of [manifolds](@ref "(Matrix) Manifolds"). Here we first state the theorem and then give a proof. - -```@eval -Main.theorem(raw"Consider a vector-valued differentiable function ``F:\mathbb{R}^N\to\mathbb{R}^N`` and assume its Jacobian is non-degenerate at a point ``x\in\mathbb{R}^N``. Then there exists a neighborhood ``U`` that contains ``F(x)`` and on which ``F`` is invertible, i.e. ``\exists{}H:U\to\mathbb{R}^N`` s.t. ``\forall{}y\in{}U,\,F\circ{}H(y) = y`` and ``H`` is differentiable."; name = "Inverse function theorem") -``` - -```@eval -Main.proof(raw"Consider a mapping ``F:\mathbb{R}^N\to\mathbb{R}^N`` and assume its Jacobian has full rank at point ``x``, i.e. ``\det{}F'(x)\neq0``. We further assume that ``F(x) = 0``, ``F'(x) = \mathbb{I}`` and ``x = 0``. Now consider a ball around ``x`` whose radius ``r`` we do not yet fix and two points ``y`` and ``z`` in that ball: ``y,z\in{}B(r)``. We further introduce the function ``G(y):=y-F(y)``. By the *mean value theorem* we have -" * Main.indentation * raw"```math -" * Main.indentation * raw"|G(y)| = |G(y) - x| = |G(y) - G(x)|\leq|y-x|\sup_{0 n = \mathrm{dim}(\mathcal{N})`` tangent mapping ``T_x\mathcal{R}`` has full rank at every point ``x\in\mathcal{N}``. Then ``\mathcal{R}(\mathcal{N})`` is a manifold *immersed* in ``\mathcal{M}``."; name = "Immersion Theorem") -``` - -The proof is again based on the [inverse function theorem](@ref "The Inverse Function Theorem"). - -```@eval -Main.proof(raw"Consider a point ``x\in\mathcal{N},`` a coordinate chart ``\varphi`` around ``x`` and a coordinate chart ``\psi`` around ``f(x).`` We now define the function -" * Main.indentation * raw"```math -" * Main.indentation * raw" F:(x_1, \ldots, x_N) \mapsto (\psi\circ{}f\circ\varphi^{-1}(x_1, \ldots, x_n), x_{n+1}, \ldots, x_N). -" * Main.indentation * raw"``` -" * Main.indentation * raw"By the inverse function theorem we can find an inverse of ``F`` for a neighborhood around the point ``(x_1, \ldots, x_n, 0, \ldots, 0)\in\mathbb{R}^N.`` We call this neighborhood ``V = V_1\times{}V_2`` and the inverse ``H.`` We now constrain ``V`` to the set ``V_1\times{}0``, which is isomorphic to a neighborhood around ``x`` in ``\mathbb{R}^n``. We then have in this neighborhood: -" * Main.indentation * raw"```math -" * Main.indentation * raw" H(\psi\circ{}f\circ\varphi^{-1}(x_1, \ldots, x_n), 0, \ldots, 0) = (x_1, \ldots, x_n, 0, \ldots, 0), -" * Main.indentation * raw"``` -" * Main.indentation * raw"And we can take -" * Main.indentation * raw"```math -" * Main.indentation * raw" y \mapsto \pi\circ{}H(\psi(y), 0 \ldots, 0) -" * Main.indentation * raw"``` -" * Main.indentation * raw"as our coordinate chart. ``\pi:\mathbb{R}^N\to\mathbb{R}^n`` is the projection onto the first ``n`` coordinates.") -``` - -We will use the immersion theorem when discussing the [symplectic solution manifold](@ref "The Symplectic Solution Manifold"). - -## Tangent Spaces - -We already alluded to tangent spaces when talking about the preimage and the immersion theorems. Here we will give a precise definition. A tangent space can be seen as the *collection of all possible velocities a curve can take at a point on a manifold*. For this consider a manifold ``\mathcal{M}`` and a point ``x`` on it and the collection of ``C^\infty`` curves through ``x``: - -```@eval -Main.definition(raw"A mapping ``\gamma:(-\epsilon, \epsilon)\to\mathcal{M}`` that is ``C^\infty`` and for which we have ``\gamma(0) = x`` is called a **``C^\infty`` curve through ``x``**.") -``` - -The tangent space of ``\mathcal{M}`` at ``x`` is the collection of the first derivatives of all ``\gamma``: - -```@eval -Main.definition(raw"The **tangent space** of ``\mathcal{M}`` at ``x`` is the collection of all ``C^\infty`` curves at ``x`` modulo the equivalence class ``\gamma_1 \sim \gamma_2 \iff \gamma_1'(0) = \gamma_2'(0)``. It is denoted by ``T_x\mathcal{M}``.") -``` - -As is customary we write ``[\gamma]`` for the equivalence class of ``\gamma`` and this is by definition equivalent to ``\gamma'(0)``. -The tangent space ``T_x\mathcal{M}`` can be shown to be homeomorphic[^3] to ``\mathbb{R}^n`` where ``n`` is the dimension of the manifold ``\mathcal{M}``. If the homeomorphism is constructed through the coordinate chart ``(\varphi, U)`` we call it ``\varphi'(x)`` or simply[^4] ``\varphi'``. If we are given a map ``g:\mathcal{M}\to\mathcal{N}`` we further define ``T_xg = (\varphi')^{-1}\circ(\varphi\circ{}g\circ\psi^{-1})'\circ{}\psi'``, i.e. a smooth map between two manifolds ``\mathcal{M}`` and ``\mathcal{N}`` induces a smooth map between the tangent spaces ``T_x\mathcal{M}`` and ``T_{g(x)}\mathcal{N}``. - -[^3]: Note that we have not formally defined addition for ``T_x\mathcal{M}``. This can be done through the definition ``[\gamma] + [\beta] = [\alpha]`` where ``\alpha`` is any ``C^\infty`` curve through ``x`` that satisfies ``\alpha'(0) = \beta(0) + \gamma(0)``. Note that we can always find such an ``\alpha`` by the [existence and uniqueness theorem](@ref "The Existence-And-Uniqueness Theorem"). - -[^4]: We will further discuss this when we introduce the [tangent bundle](@ref "The Tangent Bundle"). - -We want to demonstrate this principle of constructing the tangent space from curves through the example of ``S^2``. We consider the following curves: -1. ``\gamma_1(t) = \begin{pmatrix} 0 \\ \sin(t) \\ \cos(t) \end{pmatrix},`` -2. ``\gamma_2(t) = \begin{pmatrix} \sin(t) \\ 0 \\ \cos(t) \end{pmatrix},`` -3. ``\gamma_3(t) = \begin{pmatrix} \exp(-t ^ 2 / 2) t \sin(t) \\ \exp(-t ^ 2 / 2) t \cos(t) \\ \sqrt{1 - (t ^ 2) \exp(-t^2)} \end{pmatrix}. `` - -We now plot the manifold ``S^2``, the three curves described above and the associated tangent vectors (visualized as arrows). Note that the tangent vectors induced by ``\gamma_1`` and ``\gamma_3`` are the same; for these curves we have ``\gamma_1 \sim \gamma_3`` and the tangent vectors of those two curves coincide: - -```@eval -using CairoMakie -using ForwardDiff -using LaTeXStrings - -function plot_curve!(ax, gamma::Function; epsilon_range::T = 1.4, epsilon_spacing::T = .01, kwargs...) where T - curve_domain = -epsilon_range : epsilon_spacing : epsilon_range - curve = zeros(T, 3, length(curve_domain)) - for (i, t) in zip(axes(curve_domain, 1), curve_domain) - curve[:, i] .= gamma(t) - end - lines!(ax, curve[1, :], curve[2, :], curve[3, :]; kwargs...) -end - -function plot_arrow!(ax, gamma::Function; kwargs...) - arrow_val = ForwardDiff.derivative(gamma, 0.) - - gamma_vec = ([gamma(0)[1]], [gamma(0)[2]], [gamma(0)[3]]) - gamma_deriv_vec = ([arrow_val[1]], [arrow_val[2]], [arrow_val[3]]) - - arrows!(ax, gamma_vec..., gamma_deriv_vec...; kwargs...) -end - -function tangent_space(; n = 100) - xs = LinRange(-1.2, 1.2, n) - ys = LinRange(-1.2, 1.2, n) - zs = [one(x) * one(y) for x in xs, y in ys] - xs, ys, zs -end - -gamma_1(t) = [zero(t), sin(t), cos(t)] -gamma_2(t) = [sin(t), zero(t), cos(t)] -gamma_3(t) = [exp(-t ^ 2 / 2) * (t ^ 1) * sin(t), exp(-t ^ 2 / 2) * (t ^ 1) * cos(t), sqrt(1 - (t ^ 2) * exp(-t^2))] - -curves = (gamma_1, gamma_2, gamma_3) - -morange = RGBf(255 / 256, 127 / 256, 14 / 256) -mblue = RGBf(31 / 256, 119 / 256, 180 / 256) -mred = RGBf(214 / 256, 39 / 256, 40 / 256) -mpurple = RGBf(148 / 256, 103 / 256, 189 / 256) -mgreen = RGBf(44 / 256, 160 / 256, 44 / 256) - -colors = (morange, mblue, mred) - -function make_plot(; theme = :light) - text_color = theme == :light ? :black : :white - - fig = Figure(; backgroundcolor = :transparent) - - ax = Axis3(fig[1, 1]; - backgroundcolor = :transparent, - aspect = (1., 1., 0.8), - azimuth = π / 6, - elevation = π / 8, - xlabel = L"x_1", - ylabel = L"x_2", - zlabel = L"x_3", - xlabelcolor = text_color, - ylabelcolor = text_color, - zlabelcolor = text_color, - ) - - surface!(Main.sphere(1., [0., 0., 0.])...; alpha = .6) - - for (i, curve, color) in zip(1:length(curves), curves, colors) - plot_curve!(ax, curve; label = rich("γ", subscript(string(i)); color = text_color, font = :italic), linewidth = 2, color = color) - end - - surface!(ax, tangent_space()...; alpha = .2) - text!(.9, -.9, 1.; text = L"T_x\mathcal{M}", color = text_color) - - for (i, curve, color) in zip(1:length(curves), curves, colors) - plot_arrow!(ax, curve; linewidth = .03, color = color) - end - - axislegend(; position = (.82, .75), backgroundcolor = :transparent, color = text_color) - - fig, ax -end - -px_per_unit = Main.output_type == :html ? 1.5 : 2 -CairoMakie.save("tangent_space_light.png", make_plot(; theme = :light)[1]; px_per_unit = px_per_unit) -CairoMakie.save("tangent_space_dark.png", make_plot(; theme = :dark )[1]; px_per_unit = px_per_unit) - -nothing -``` - -![Visualization of how the tangent space is constructed.](tangent_space_light.png) -![Visualization of how the tangent space is constructed.](tangent_space_dark.png) - -The tangent space ``T_x\mathcal{M}`` for - -```math -x = \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix} -``` - - is also shown. - -## Vector Fields - -A time-independent vector field[^5] is an object that specifies a velocity for every point on a domain. We first give the definition of a vector field on the vector space ``\mathbb{R}^n`` and limit ourselves here to ``C^\infty`` vector fields: - -[^5]: Also called *ordinary differential equation* (ODE). - -```@eval -Main.definition(raw"A **vector field** on ``\mathbb{R}^n`` is a smooth map ``X:\mathbb{R}^n\to\mathbb{R}^n``.") -``` - -The definition of a vector field on a manifold is not much more complicated: - -```@eval -Main.definition(raw"A **vector field** on ``\mathcal{M}`` is a map ``X`` defined on ``\mathcal{M}`` such that ``X(x)\in{}T_x\mathcal{M}`` and ``\varphi'\circ{}X\circ(\varphi)^{-1}`` is smooth for any coordinate chart ``(\varphi, U)`` that contains ``x``.") -``` - -In the section on the [existence-and-uniqueness theorem](@ref "The Existence-And-Uniqueness Theorem") we show that every vector field has a unique solution given an initial condition; i.e. given a point ``x\in\mathcal{M}`` and a vector field ``X`` we can find a curve ``\gamma`` such that ``\gamma(0) = x`` and ``\gamma'(t) = X(\gamma(t))`` for all ``t`` in some interval ``(-\epsilon, \epsilon)``. - - -## The Tangent Bundle - -To each manifold ``\mathcal{M}`` we can associate another manifold which we call the *tangent bundle* and denote by ``T\mathcal{M}``. The points on this manifold are: - -```math -T\mathcal{M} = \{ (x, v_x): x\in\mathcal{M},\, v_x\in{}T_x\mathcal{M} \}. -``` - -Coordinate charts on this manifold can be constructed in a straightforward manner; for every coordinate chart ``\varphi_U`` the map ``\varphi_U'(x)`` gives a homeomorphism between ``T_x\mathcal{M}`` and ``\mathbb{R}^n`` for any ``x\in{}U``. We can then find a neighborhood of any point ``(x, v_x)`` by taking ``\pi^{-1}(U) = \{(x, v_x): x\in{}U, v_x\in{}T_x\mathcal{M}\}`` and this neighborhood is isomorphic to ``\mathbb{R}^{2n}`` via ``(x, v_x) \mapsto (\varphi_U(x), \varphi'(x)v_x)``. The [geodesic spray](@ref "Geodesic Sprays and the Exponential Map") is an important vector field defined on ``T\mathcal{M}``. - -## Library Functions - -```@docs -Manifold -rand(::Type{MT}, ::Integer, ::Integer) where MT <: Manifold -rand(::GeometricMachineLearning.Backend, ::Type{MT}, ::Integer, ::Integer) where MT <: Manifold -``` - -```@raw latex -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -absil2008optimization -``` - -```@raw latex -\end{comment} -``` \ No newline at end of file diff --git a/docs/src/manifolds/metric_and_vector_spaces.md b/docs/src/manifolds/metric_and_vector_spaces.md deleted file mode 100644 index f01a8e4da..000000000 --- a/docs/src/manifolds/metric_and_vector_spaces.md +++ /dev/null @@ -1,115 +0,0 @@ -# (Topological) Metric Spaces - -A metric space is a certain class of a topological space where the topology is *induced through a metric*. We define this notion now: - -```@eval -Main.definition(raw"A **metric** on a topological space ``\mathcal{M}`` is a mapping ``d:\mathcal{M}\times\mathcal{M}\to\mathbb{R}`` such that the following three conditions hold: -" * -Main.indentation * raw"1. ``d(x, y) = 0 \iff x = y`` for every ``x,y\in\mathcal{M}``, i.e. the distance between two points is zero if and only if they are the same, -" * -Main.indentation * raw"2. ``d(x, y) = d(y, x)``, -" * -Main.indentation * raw"3. ``d(x, z) \leq d(x, y) + d(y, z)``. -" * -Main.indentation * raw"The second condition is referred to as *symmetry* and the third condition is referred to as the *triangle inequality*.") -``` - -We give some examples of metric spaces that are relevant for us: - -```@eval -Main.example(raw"The real line ``\mathbb{R}`` with the metric defined by the absolute distance between two points: ``d(x, y) = |y - x|``.") -``` - -```@eval -Main.example(raw"The vector space ``\mathbb{R}^n`` with the *Euclidean distance* ``d_2(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}``.") -``` - -```@eval -Main.example(raw"The space of continuous functions ``\mathcal{C} = \{f:(-\epsilon, \epsilon)\to\mathbb{R}^n\}`` with the metric ``d_\infty(f_1, f_2) = \mathrm{sup}_{t\in(-\epsilon, \epsilon)}|f_1(t) - f_2(t)|.``") -``` - -```@eval -Main.proof(raw"We have to show the triangle inequality: -" * -Main.indentation * raw"```math -" * -Main.indentation * raw"\begin{aligned} -" * -Main.indentation * raw"d_\infty(d_1, d_3) = \mathrm{sup}_{t\in(-\epsilon, \epsilon)}|f_1(t) - f_3(t)| & \leq \mathrm{sup}_{t\in(-\epsilon, \epsilon)}(|f_1(t) - f_2(t)| + |f_2(t) - f_3(t)|) \\ -" * -Main.indentation * raw"& \leq \mathrm{sup}_{t\in(-\epsilon, \epsilon)}|f_1(t) - f_2(t)| + \mathrm{sup}_{t\in(-\epsilon, \epsilon)}|f_1(t) - f_2(t)|. -" * -Main.indentation * raw"\end{aligned} -" * -Main.indentation * raw"``` -" * -Main.indentation * raw"This shows that ``d_\infty`` is indeed a metric.") -``` - -```@eval -Main.example(raw"Any Riemannian manifold is a metric space.") -``` - -This last example shows that *metric spaces need not be vector spaces*, i.e. spaces for which we can define a metric but not addition of two elements. This will be discussed in more detail in the section on [Riemannian manifolds](@ref "Riemannian Manifolds"). - -## Complete Metric Spaces - -To define *complete metric spaces* we first need the definition of a *Cauchy sequence*. - -```@eval -Main.definition(raw"A **Cauchy sequence** is a sequence ``(a_n)_{n\in\mathbb{N}}`` for which, given any ``\epsilon>0``, we can find an integer ``N`` such that ``d(a_n, a_m) < \epsilon`` for all ``n, m \geq N``.") -``` - -Now we can give the definition of a *complete metric space*: - -```@eval -Main.definition(raw"A **complete metric space** is one for which every Cauchy sequence converges.") -``` - -Completeness of the real numbers is most often seen as an axiom and therefore stated without proof. This also implies completeness of ``\mathbb{R}^n`` [lang2012real](@cite). - - -# (Topological) Vector Spaces - -Vector Spaces are, like metric spaces, topological spaces which we endow with additional structure. - -```@eval -Main.definition(raw"A **vector space** ``\mathcal{V}`` is a topological space for which we define an operation called *addition* and denoted by ``+`` and an operation called *scalar multiplication* (by elements of ``\mathbb{R}``) denoted by ``x \mapsto ax`` for ``x\in\mathcal{V}`` and ``x\in\mathbb{R}`` for which the following hold for all ``x, y, z\in\mathcal{V}`` and ``a, b\in\mathbb{R}``: -" * -Main.indentation * raw"1. ``x + (y + z) = (x + y) + z,`` -" * -Main.indentation * raw"2. ``x + y = y + x,`` -" * -Main.indentation * raw"3. ``\exists 0 \in \mathcal{V}\text{such that }x + 0 = x,`` -" * -Main.indentation * raw"4. ``\exists -x \in \mathcal{V}\text{ such that }x + (-x) = 0,`` -" * -Main.indentation * raw"5. ``a(ax) = (ab)x,`` -" * -Main.indentation * raw"6. ``1x = x`` for ``1\in\mathbb{R},`` -" * -Main.indentation * raw"7. ``a(x + y) = ax + ay,`` -" * -Main.indentation * raw"8. ``(a + b)x = ax + bx.`` -" * -Main.indentation * raw"The first law is known as *associativity*, the second one as *commutativity* and the last two ones are known as *distributivity*.") -``` - -The topological spaces ``\mathbb{R}`` and ``\mathbb{R}^{n}`` are (almost) trivially vector spaces. The same is true for many function spaces. One of the special aspects of `GeometricMachineLearning` is that it can deal with spaces that are not vector spaces, but manifolds. All vector spaces are however manifolds. - -```@raw latex -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -lang2012real -``` - -```@raw latex -\end{comment} -``` \ No newline at end of file diff --git a/docs/src/manifolds/riemannian_manifolds.md b/docs/src/manifolds/riemannian_manifolds.md deleted file mode 100644 index 17b6e9ddf..000000000 --- a/docs/src/manifolds/riemannian_manifolds.md +++ /dev/null @@ -1,230 +0,0 @@ -# Riemannian Manifolds - -A Riemannian manifold is a manifold ``\mathcal{M}`` that we endow with a mapping ``g`` that smoothly[^1] assigns a [metric](@ref "(Topological) Metric Spaces") ``g_x`` to each tangent space ``T_x\mathcal{M}``. By a slight abuse of notation we will also refer to this ``g`` as a *metric*. - -[^1]: Smooth here refers to the fact that ``g:\mathcal{M}\to\text{(Space of Metrics)}`` has to be a smooth map. But in order to discuss this in detail we would have to define a topology on the space of metrics. A more detailed discussion can be found in [lang2012fundamentals, bishop1980tensor, do1992riemannian](@cite). - -After having defined a metric ``g`` we can *associate a length* to each curve ``\gamma:[0, t] \to \mathcal{M}`` through: - -```math -L(\gamma) = \int_0^t \sqrt{g_{\gamma(s)}(\gamma'(s), \gamma'(s))}ds. -``` - -This ``L`` turns ``\mathcal{M}`` into a metric space: - -```@eval -Main.definition(raw"The **metric on a Riemannian manifold** ``\mathcal{M}`` is -" * -Main.indentation * raw"```math -" * -Main.indentation * raw"d(x, y) = \inf_{\substack{\text{$\gamma(0) = x$ and}\\ - \gamma(t) = y}}L(\gamma), -" * -Main.indentation * raw"``` -" * -Main.indentation * raw"where ``t`` can be chosen arbitrarily.") -``` - -If a curve is minimal with respect to the function ``L`` we call it the *shortest curve* or a geodesic. So we say that a curve ``\gamma:[0, t]\to\mathcal{M}`` is a geodesic if there is no shorter curve that can connect two points in ``\gamma([0, t])``, i.e. - -```math -d(\gamma(t_i), \gamma(t_f)) = \int_{t_i}^{t_f}\sqrt{g_{\gamma(s)}(\gamma'(s), \gamma'(s))}ds, -``` -for any ``t_i, t_f\in[0, t]``. - -An important result of Riemannian geometry states that there exists a vector field ``X`` on ``T\mathcal{M}``, called the *geodesic spray*, whose integral curves are derivatives of geodesics. We formalize this statement as a theorem in the next section. - - -## Geodesic Sprays and the Exponential Map - -To every Riemannian manifold we can naturally associate a vector field called the *geodesic spray* or *geodesic equation*. For our purposes it is enough to state that this vector field is unique and well-defined [do1992riemannian](@cite). - -The important property of the geodesic spray is - -```@eval -Main.theorem(raw"Given an initial point ``x`` and an initial velocity ``v_x``, an integral curve for the geodesic spray is of the form ``t \mapsto (\gamma_{v_x}(t), \gamma_{v_x}'(t))`` where ``\gamma_{v_x}`` is a geodesic. We further have the property that the integral curve for the geodesic spray for an initial point ``x`` and an initial velocity ``\eta\cdot{}v_x`` (where ``\eta`` is a scalar) is of the form ``t \mapsto (\gamma_{\eta\cdot{}v_x}(t), \gamma_{\eta\cdot{}v_x}'(t)) = (\gamma_{v_x}(\eta\cdot{}t), \eta\cdot\gamma_{v_x}'(\eta\cdot{}t)).``") -``` - -It is therefore customary to introduce the *exponential map* ``\exp:T_x\mathcal{M}\to\mathcal{M}`` as - -```math -\exp(v_x) := \gamma_{v_x}(1), -``` - -and we see that ``\gamma_{v_x}(t) = \exp(t\cdot{}v_x)``. In `GeometricMachineLearning` we denote the exponential map by [`geodesic`](@ref geodesic(::StiefelManifold{T}, ::AbstractMatrix{T}) where T) to avoid confusion with the matrix exponential map[^2] which is called as `exp` in `Julia`. So we use the definition: - -[^2]: The Riemannian exponential map and the matrix exponential map coincide for many matrix Lie groups. - -```math - \mathtt{geodesic}(x, v_x) \equiv \exp(v_x). -``` - -We give an example of using this function here: - -```@setup s2_retraction -using CairoMakie -``` - -```@example s2_retraction -using GeometricMachineLearning # hide -import Random # hide -Random.seed!(123) # hide - -Y = rand(StiefelManifold, 3, 1) - -v = 2 * rand(3, 1) -Δ = v - Y * (v' * Y) - -morange = RGBf(255 / 256, 127 / 256, 14 / 256) # hide -mred = RGBf(214 / 256, 39 / 256, 40 / 256) # hide -function set_up_plot(; theme = :dark) # hide -text_color = theme == :dark ? :white : :black # hide -fig = Figure(; backgroundcolor = :transparent, size = (900, 675)) # hide -ax = Axis3(fig[1, 1]; # hide - backgroundcolor = (:tomato, .5), # hide - aspect = (1., 1., 1.), # hide - xlabel = L"x_1", # hide - ylabel = L"x_2", # hide - zlabel = L"x_3", # hide - xgridcolor = text_color, # hide - ygridcolor = text_color, # hide - zgridcolor = text_color, # hide - xtickcolor = text_color, # hide - ytickcolor = text_color, # hide - ztickcolor = text_color, # hide - xlabelcolor = text_color, # hide - ylabelcolor = text_color, # hide - zlabelcolor = text_color, # hide - xypanelcolor = :transparent, # hide - xzpanelcolor = :transparent, # hide - yzpanelcolor = :transparent, # hide - limits = ([-1, 1], [-1, 1], [-1, 1]), # hide - azimuth = π / 7, # hide - elevation = π / 7, # hide - # height = 75., # hide - ) # hide -# plot a sphere with radius one and origin 0 -surface!(ax, Main.sphere(1., [0., 0., 0.])...; alpha = .5, transparency = true) - -point_vec = ([Y[1]], [Y[2]], [Y[3]]) -scatter!(ax, point_vec...; color = morange, marker = :star5, markersize = 30) - -arrow_vec = ([Δ[1]], [Δ[2]], [Δ[3]]) -arrows!(ax, point_vec..., arrow_vec...; color = mred, linewidth = .02) - -fig, ax # hide -end # hide - -fig_light = set_up_plot(; theme = :light)[1] # hide -fig_dark = set_up_plot(; theme = :dark)[1] # hide - -CairoMakie.save("sphere_with_tangent_vec_light.png", fig_light; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide -CairoMakie.save("sphere_with_tangent_vec_dark.png", fig_dark; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide - -nothing # hide -``` - -![A tangent vector on ``\mathcal{M}`` determines a direction.](sphere_with_tangent_vec_light.png) -![A tangent vector on ``\mathcal{M}`` determines a direction.](sphere_with_tangent_vec_dark.png) - - -We now solve the geodesic spray for ``\eta\cdot\Delta`` for ``\eta = 0.1, 0.2, \ldots, 5.5`` with the function [`geodesic`](@ref geodesic(::StiefelManifold{T}, ::AbstractMatrix{T}) where T) and plot the corresponding points: - -```@example s2_retraction -Δ_increments = [Δ * η for η in 0.1 : 0.1 : 5.5] - -Y_increments = [geodesic(Y, Δ_increment) for Δ_increment in Δ_increments] - -function make_plot_with_solution(; theme = :dark) # hide -fig, ax = set_up_plot(; theme = theme) # hide -for Y_increment in Y_increments - scatter!(ax, [Y_increment[1]], [Y_increment[2]], [Y_increment[3]]; - color = mred) -end - -fig # hide -end # hide - -fig_light = make_plot_with_solution(; theme = :light) # hide -fig_dark = make_plot_with_solution(; theme = :dark) # hide - -CairoMakie.save("sphere_with_tangent_vec_and_geodesic_light.png", fig_light; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide -CairoMakie.save("sphere_with_tangent_vec_and_geodesic_dark.png", fig_dark; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide - -nothing # hide -``` - -![Solving the geodesic gives a path along the tangent vector, equivalent to a straight line in flat space.](sphere_with_tangent_vec_and_geodesic_light.png) -![Solving the geodesic gives a path along the tangent vector, equivalent to a straight line in flat space.](sphere_with_tangent_vec_and_geodesic_dark.png) - -A geodesic can be seen as the *equivalent of a straight line* on a manifold. Also note that we drew a random element form [`StiefelManifold`](@ref) here, and not from ``S^2``. This is because the category of [Stiefel manifolds](@ref "The Stiefel Manifold") is more general than the category of spheres ``S^n``: ``St(1, 3) \simeq S^2``. - -## The Riemannian Gradient - -The *Riemannian gradient* is essential when talking about optimization on manifolds. - -```@eval -Main.definition(raw"The Riemannian gradient of a function ``L:\mathcal{M}\to\mathbb{R}`` is a vector field ``\mathrm{grad}^gL`` (or simply ``\mathrm{grad}L``) for which we have -" * Main.indentation * raw"```math -" * Main.indentation * raw" g_x(\mathrm{grad}^gL(x), v_x) = (\nabla_{\varphi_U(x)}(L\circ\varphi_U^{-1}))^T \varphi_U'(v_x), -" * Main.indentation * raw"``` -" * Main.indentation * raw"for all ``v_x\in{}T_x\mathcal{M}.`` In the expression above ``\varphi_U`` is some coordinate chart defined in a neighborhood ``U`` around ``x``.") -``` - -In the definition above ``\nabla`` indicates the *Euclidean gradient*: -```math - \nabla_xf = \begin{pmatrix} \frac{\partial{}f}{\partial{}x_1} \\ \cdots \\ \frac{\partial{}f}{\partial{}x_n} \end{pmatrix}. -``` - -We can also describe the Riemannian gradient through differential curves: - -```@eval -Main.definition(raw"The Riemannian gradient of ``L`` is a vector field ``\mathrm{grad}^gL`` for which -" * Main.indentation * raw"```math -" * Main.indentation * raw"g_x(\mathrm{grad}^gL(x), \dot{\gamma}(0)) = \frac{d}{dt}L(\gamma(t)), -" * Main.indentation * raw"``` -" * Main.indentation * raw"where ``\gamma`` is a ``C^\infty`` curve through ``x``.") -``` - -By the *non degeneracy* of ``g`` the Riemannian gradient always exists [bishop1980tensor](@cite). In the following we will also write ``\mathrm{grad}^gL(x) = \mathrm{grad}^g_xL = \mathrm{grad}_xL.`` We will give specific examples of this when discussing the [Stiefel manifold](@ref "The Stiefel Manifold") and the [Grassmann manifold](@ref "The Grassmann Manifold"). - - -## Gradient Flows and Riemannian Optimization - -In `GeometricMachineLearning` we can include weights in neural networks that are part of a manifold. Training such neural networks amounts to *Riemannian optimization* and hence solving the *gradient flow* equation. The gradient flow equation is given by - -```math -X(x) = - \mathrm{grad}_xL. -``` - -Solving this gradient flow equation will then lead us to a local minimum on ``\mathcal{M}``. This will be elaborated on when talking about [optimizers](@ref "Neural Network Optimizers"). In practice we cannot solve the gradient flow equation directly and have to rely on approximations. The most straightforward approximation (and one that serves as a basis for all the optimization algorithms in `GeometricMachineLearning`) is to take the point ``(x, X(x))`` as an initial condition for the geodesic spray and then solve the ODE for a small time step. Such an update rule, i.e. - -```math -x^{(t)} \leftarrow \gamma_{X(x^{(t-1)})}(\Delta{}t)\text{ with $\Delta{}t$ the time step}, -``` - -we call the *gradient optimization scheme*. - -## Library Functions - -```@docs -geodesic(::StiefelManifold{T}, ::AbstractMatrix{T}) where T -``` - -```@raw latex -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -lang2012fundamentals -do1992riemannian -``` - -```@raw latex -\end{comment} -``` \ No newline at end of file diff --git a/docs/src/optimizers/manifold_related/parallel_transport.md b/docs/src/optimizers/manifold_related/parallel_transport.md deleted file mode 100644 index f520342b7..000000000 --- a/docs/src/optimizers/manifold_related/parallel_transport.md +++ /dev/null @@ -1,232 +0,0 @@ -# Parallel Transport - -The concept of *parallel transport along a geodesic* ``\gamma:[0, T]\to\mathcal{M}`` describes moving a tangent vector from ``T_x\mathcal{M}`` to ``T_{\gamma(t)}\mathcal{M}`` such that its orientation with respect to the geodesic is preserved. - -A precise definition of parallel transport needs a notion of a *connection* [lang2012fundamentals, bishop1980tensor, bendokat2020grassmann](@cite) and we forego it here. We simply state how to parallel transport vectors on the Lie group ``SO(N)`` and the homogeneous spaces ``St(n, N)`` and ``Gr(n, N)``. - -```@eval -Main.theorem(raw"Given two elements ``B^A_1, B^A_2\in{}T_AG`` the parallel transport of ``B^A_2`` along the geodesic of ``B^A_1`` is given by -" * Main.indentation * raw"```math -" * Main.indentation * raw"\Pi_{A\to\gamma_{B^A_1}(t)}B^A_2 = A\exp(t\cdot{}A^{-1}B^A_1)A^{-1}B^A_2 = A\exp(t\cdot{}B_1)B_2, -" * Main.indentation * raw"``` -" * Main.indentation * raw"where ``B_i := A^{-1}B^A_i.``") -``` - -For the Stiefel manifold this is not much more complicated[^1]: - -[^1]: Here we do not provide a detailed proof that this constitutes a sound expression from the perspective of Riemannian geometry. A proof can be found in [schlarb2024covariant](@cite). - -```@eval -Main.theorem(raw"Given two elements ``\Delta_1, \Delta_2\in{}T_Y\mathcal{M}``, the parallel transport of ``\Delta_2`` along the geodesic of ``\Delta_1`` is given by -" * Main.indentation * raw"```math -" * Main.indentation * raw"\Pi_{Y\to\gamma_{\Delta_1}(t)}\Delta_2 = \exp(t\cdot\Omega(Y, \Delta_1))\Delta_2 = \lambda(Y)\exp(\bar{B}_1)\lambda(Y)^{-1}\Delta_2, -" * Main.indentation * raw"``` -" * Main.indentation * raw"where ``\bar{B}_1 = \lambda(Y)^{-1}\Omega(Y, \Delta_1)\lambda(Y).``") -``` - -We can further modify the expression of parallel transport for the Stiefel manifold: - -```math -\Pi_{Y\to\gamma_{\Delta_1}(t)}\Delta_2 = \lambda(Y)\exp(B_1)\lambda(Y)\Omega(Y, \Delta_2)Y = \lambda(Y)\exp(B_1)B_2E, -``` - -where ``B_2 = \lambda(Y)^{-1}\Omega(Y, \Delta_2)\lambda(Y).`` We can now define explicit updating rules for the [global section](@ref "Global Sections") ``\Lambda^{(\cdot)}``, the element of the homogeneous space ``Y^{(\cdot)}``, the tangent vector ``\Delta^{(\cdot)}`` and ``D^{(\cdot)} = (\Lambda^{(\cdot)})^{-1}\Omega(\Delta^{(\cdot)})\Lambda^{(\cdot)}``, its representation in ``\mathfrak{g}^\mathrm{hor}``. - -We thus have: -1. ``\Lambda^{(t)} \leftarrow \Lambda^{(t-1)}\exp(B^{(t-1)}),`` -2. ``Y^{(t)} \leftarrow \Lambda^{(t)}E,`` -3. ``\Delta^{(t)} \leftarrow \Lambda^{(t-1)}\exp(B^{(t-1)})(\Lambda^{(t-1)})^{-1}\Delta^{(t-1)} = \Lambda^{(t)}D^{(t-1)}E,`` -4. ``D^{(t)} \leftarrow D^{(t-1)}.`` - -So we conveniently take parallel transport of vectors into account by representing them in ``\mathfrak{g}^\mathrm{hor}``: ``D`` does not change. - -To demonstrate parallel transport we again use the example from when we introduced the concept of [geodesics](@ref "Geodesic Sprays and the Exponential Map"). We first set up the problem: - -```@setup s2_parallel_transport -using CairoMakie -``` - -```@setup s2_parallel_transport -using GeometricMachineLearning -import Random # hide -Random.seed!(123) # hide - -Y = rand(StiefelManifold, 3, 1) -# needed because we will change `Y` later on -Y_copy = StiefelManifold(copy(Y.A)) - -v = 2 * rand(3, 1) -v₂ = 1 * rand(3, 1) -Δ = rgrad(Y, v) -Δ₂ = rgrad(Y, v₂) - -morange = RGBf(255 / 256, 127 / 256, 14 / 256) # hide -mred = RGBf(214 / 256, 39 / 256, 40 / 256) # hide -mpurple = RGBf(148 / 256, 103 / 256, 189 / 256) # hide - -function set_up_plot(; theme = :dark) # hide -fig = Figure(; backgroundcolor = :transparent, size = (900, 675)) # hide -text_color = theme == :dark ? :white : :black # hide -ax = Axis3(fig[1, 1]; # hide - backgroundcolor = (:tomato, .5), # hide - aspect = (1., 1., 1.), # hide - xlabel = L"x_1", # hide - ylabel = L"x_2", # hide - zlabel = L"x_3", # hide - xgridcolor = text_color, # hide - ygridcolor = text_color, # hide - zgridcolor = text_color, # hide - xtickcolor = text_color, # hide - ytickcolor = text_color, # hide - ztickcolor = text_color, # hide - xlabelcolor = text_color, # hide - ylabelcolor = text_color, # hide - zlabelcolor = text_color, # hide - xypanelcolor = :transparent, # hide - xzpanelcolor = :transparent, # hide - yzpanelcolor = :transparent, # hide - limits = ([-1, 1], [-1, 1], [-1, 1]), - azimuth = π / 7, # hide - elevation = π / 7, # hide - # height = 75., - ) # hide - -# plot a sphere with radius one and origin 0 -surface!(ax, Main.sphere(1., [0., 0., 0.])...; alpha = .5, transparency = true) - -point_vec = ([Y_copy[1]], [Y_copy[2]], [Y_copy[3]]) -scatter!(ax, point_vec...; color = morange, marker = :star5, markersize = 30) - -arrow_vec = ([Δ[1]], [Δ[2]], [Δ[3]]) -arrows!(ax, point_vec..., arrow_vec...; color = mred, linewidth = .02) - -arrow_vec2 = ([Δ₂[1]], [Δ₂[2]], [Δ₂[3]]) -arrows!(ax, point_vec..., arrow_vec2...; color = mpurple, linewidth = .02) - -fig, ax # hide -end # hide - -fig_light = set_up_plot(; theme = :light)[1] -fig_dark = set_up_plot(; theme = :dark)[1] -CairoMakie.save("two_vectors_light.png", fig_light; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide -CairoMakie.save("two_vectors_dark.png", fig_dark; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide - -nothing # hide -``` - -![The purple vector should be transported along the geodesic of the red one.](two_vectors_light.png) -![The purple vector should be transported along the geodesic of the red one.](two_vectors_dark.png) - -Note that we have chosen the arrow here to have the same direction as before but only about half the magnitude. We further drew another arrow that we want to parallel transport (the purple arrow). - -```@example s2_parallel_transport -using GeometricOptimizers: update_section! # hide - -λY = GlobalSection(Y) -B = global_rep(λY, Δ) -B₂ = global_rep(λY, Δ₂) - -E = StiefelProjection(3, 1) -Y_increments = [] -Δ_transported = [] -Δ₂_transported = [] - -const n_steps = 6 -const timestep = 2 - -for _ in 1:n_steps - update_section!(λY, timestep * B, geodesic) - push!(Y_increments, copy(λY.Y)) - push!(Δ_transported, Matrix(λY) * B * E) - push!(Δ₂_transported, Matrix(λY) * B₂ * E) -end -nothing # hide -``` - -```@setup s2_parallel_transport -function plot_parallel_transport(; theme = :dark) # hide -fig, ax = set_up_plot(; theme = theme) # hide -for Y_increment in Y_increments - scatter!(ax, [Y_increment[1]], [Y_increment[2]], [Y_increment[3]]; - color = mred) -end - -for (color, vec_transported) in zip((mred, mpurple), (Δ_transported, Δ₂_transported)) - for (Y_increment, vec_increment) in zip(Y_increments, vec_transported) - point_vec = ([Y_increment[1]], [Y_increment[2]], [Y_increment[3]]) - arrow_vec = ([vec_increment[1]], [vec_increment[2]], [vec_increment[3]]) - arrows!(ax, point_vec..., arrow_vec...; color = color, linewidth = .02) - end -end - -fig, ax -end # hide - -fig_light, ax_light = plot_parallel_transport(; theme = :light) # hide -fig_dark, ax_dark = plot_parallel_transport(; theme = :dark) # hide -CairoMakie.save("parallel_transport_light.png", fig_light; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide -CairoMakie.save("parallel_transport_dark.png", fig_dark; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide -hidedecorations!(ax_light) # hide -hidespines!(ax_light) # hide -CairoMakie.save("parallel_transport_naked.png", fig_light; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide - -nothing # hide -``` - -![Parallel transport of the purple vector along the geodesic of the red one.](parallel_transport_light.png) -![Parallel transport of the purple vector along the geodesic of the red one.](parallel_transport_dark.png) - -Note that the angle between the two vector is preserved as we go along the geodesic. - - -```@raw latex -\section*{Chapter Summary} - -In this chapter we introduced our \textit{optimizer framework} which will be used to efficiently train symplectic autoencoders and transformers with orthogonality constraints in Part IV. We proposed extending standard neural network optimizers to homogeneous spaces by introducing the extra operations \texttt{rgrad}, \texttt{global\_rep} and ``Retraction.'' The definition of a retraction we used here was slightly different from the usual one. We defined retractions as maps from the \textit{global tangent space representation} $\mathfrak{g}^\mathrm{hor}$ to the associated Lie group (and in addition satisfy two more conditions), i.e. -\begin{equation*} - \mathrm{Retraction}: \mathfrak{g}^\mathrm{hor} \to G. -\end{equation*} - -We further discussed what the operations \texttt{rgrad}, \texttt{global\_rep} and ``Retraction'' look like in practice and concluded by introducing the concept of \textit{parallel transport}. The presentation was accompanied by code snippets that demonstrate the application interface of \texttt{GeometricMachineLearning} throughout the chapter. - -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -lang2012fundamentals -bishop1980tensor -bendokat2020grassmann -schlarb2024covariant -``` - -```@raw latex -\end{comment} -``` - -```@raw html - -``` \ No newline at end of file diff --git a/docs/src/optimizers/manifold_related/retractions.md b/docs/src/optimizers/manifold_related/retractions.md deleted file mode 100644 index 4334087d9..000000000 --- a/docs/src/optimizers/manifold_related/retractions.md +++ /dev/null @@ -1,407 +0,0 @@ -# Retractions - -In practice we usually do not solve the geodesic equation exactly in each optimization step (even though this is possible and computationally feasible), but prefer approximations that are called "retractions" [absil2008optimization](@cite) for numerical stability. The definition of a retraction in `GeometricMachineLearning` is slightly different from how it is usually defined in textbooks [absil2008optimization, hairer2006geometric](@cite). We discuss these differences here. - -## Classical Retractions - -By "classical retraction" we here mean the textbook definition. - -```@eval -Main.definition(raw"A **classical retraction** is a smooth map -" * Main.indentation * raw"```math -" * Main.indentation * raw"R: T\mathcal{M}\to\mathcal{M}:(x,v)\mapsto{}R_x(v), -" * Main.indentation * raw"``` -" * Main.indentation * raw"such that each curve ``c(t) := R_x(tv)`` is a local approximation of a geodesic, i.e. the following two conditions hold: -" * Main.indentation * raw"1. ``c(0) = x`` and -" * Main.indentation * raw"2. ``c'(0) = v.`` -") -``` - -Perhaps the most common example for matrix manifolds is the *Cayley retraction*. It is a retraction for many matrix Lie groups [hairer2006geometric, bendokat2021real, gao2021riemannian](@cite). - -```@eval -Main.example(raw"The **Cayley retraction** for ``V\in{}T_\mathbb{I}G\equiv\mathfrak{g}`` is defined as -" * Main.indentation * raw"```math -" * Main.indentation * raw"\mathrm{Cayley}(V) = \left(\mathbb{I} - \frac{1}{2}V\right)^{-1}\left(\mathbb{I} +\frac{1}{2}V\right). -" * Main.indentation * raw"```") -``` - -We show that the Cayley transform is a retraction for ``G = SO(N)`` at ``\mathbb{I}\in{}SO(N)``: -```@eval -Main.proof(raw"The Cayley transform trivially satisfies ``\mathrm{Cayley}(\mathbb{O}) = \mathbb{I}``. So what we have to show is the second condition for a retraction and that ``\mathrm{Cayley}(V)\in{}SO(N)``. For this take ``V\in\mathfrak{so}(N).`` We then have -" * Main.indentation * raw"```math -" * Main.indentation * raw"\frac{d}{dt}\bigg|_{t = 0}\mathrm{Cayley}(tV) = \frac{d}{dt}\bigg|_{t = 0}\left(\mathbb{I} - \frac{1}{2}tV\right)^{-1}\left(\mathbb{I} +\frac{1}{2}tV\right) = \frac{1}{2}V - \frac{1}{2}V^T = V, -" * Main.indentation * raw"``` -" * Main.indentation * raw"which satisfies the second condition. We further have -" * Main.indentation * raw"```math -" * Main.indentation * raw"\frac{d}{dt}\bigg|_{t = 0}(\mathrm{Cayley}(tV))^T\mathrm{Cayley}(tV) = (\frac{1}{2}V - \frac{1}{2}V^T)^T + \frac{1}{2}V - \frac{1}{2}V^T = 0. -" * Main.indentation * raw"``` -" * Main.indentation * raw"This proofs that the Cayley transform maps to ``SO(N)``.") -``` - -We should mention that the factor ``\frac{1}{2}`` is sometimes left out in the definition of the Cayley transform when used in different contexts. But it is necessary for defining a retraction as without it the second condition is not satisfied. - -```@eval -Main.remark(raw"We can also use the Cayley retraction at a different point than the identity ``\mathbb{I}.`` For this consider ``\bar{A}\in{}SO(N)`` and ``\bar{B}\in{}T_{\bar{A}}SO(N) = \{\bar{B}\in\mathbb{R}^{N\times{}N}: \bar{A}^T\bar{B} + \bar{B}^T\bar{A} = \mathbb{O}\}``. We then have ``\bar{A}^T\bar{B}\in\mathfrak{so}(N)`` and -" * Main.indentation * raw"```math -" * Main.indentation * raw" \overline{\mathrm{Cayley}}: T_{\bar{A}}SO(N) \to SO(N), \bar{B} \mapsto \bar{A}\mathrm{Cayley}(\bar{A}^T\bar{B}), -" * Main.indentation * raw"``` -" * Main.indentation * raw"is a retraction ``\forall{}\bar{A}\in{}SO(N)``.") -``` - -As a retraction is always an approximation of the geodesic map, we now compare the [`cayley`](@ref cayley(::StiefelLieAlgHorMatrix)) retraction for the example we introduced along [Riemannian manifolds](@ref "Geodesic Sprays and the Exponential Map"): - -```@setup s2_retraction -using CairoMakie -``` - -```@setup s2_retraction -using GeometricMachineLearning -import Random # hide -Random.seed!(123) # hide - -Y = rand(StiefelManifold, 3, 1) - -v = 2 * rand(3, 1) -Δ = v - Y * (v' * Y) - -function do_setup(; theme=:light) - text_color = theme == :dark ? :white : :black # hide - fig = Figure(; backgroundcolor = :transparent, size = (900, 675)) # hide - ax = Axis3(fig[1, 1]; # hide - backgroundcolor = (:tomato, .5), # hide - aspect = (1., 1., 1.), # hide - xlabel = L"x_1", # hide - ylabel = L"x_2", # hide - zlabel = L"x_3", # hide - xgridcolor = text_color, # hide - ygridcolor = text_color, # hide - zgridcolor = text_color, # hide - xtickcolor = text_color, # hide - ytickcolor = text_color, # hide - ztickcolor = text_color, # hide - xlabelcolor = text_color, # hide - ylabelcolor = text_color, # hide - zlabelcolor = text_color, # hide - xypanelcolor = :transparent, # hide - xzpanelcolor = :transparent, # hide - yzpanelcolor = :transparent, # hide - limits = ([-1, 1], [-1, 1], [-1, 1]), - azimuth = π / 7, # hide - elevation = π / 7, # hide - # height = 75., - ) # hide - - # plot a sphere with radius one and origin 0 - surface!(ax, Main.sphere(1., [0., 0., 0.])...; alpha = .5, transparency = true) - - morange = RGBf(255 / 256, 127 / 256, 14 / 256) # hide - point_vec = ([Y[1]], [Y[2]], [Y[3]]) - scatter!(ax, point_vec...; color = morange, marker = :star5, markersize = 30) - - fig, ax, point_vec -end - -mred = RGBf(214 / 256, 39 / 256, 40 / 256) # hide -mblue = RGBf(31 / 256, 119 / 256, 180 / 256) - -nothing -``` - -```@example s2_retraction -η_increments = 0.2 : 0.2 : 5.4 -Δ_increments = [Δ * η for η in η_increments] - -Y_increments_geodesic = [geodesic(Y, Δ_increment) for Δ_increment in Δ_increments] -Y_increments_cayley = [cayley(Y, Δ_increment) for Δ_increment in Δ_increments] -nothing # hide -``` - -```@setup s2_retraction -function make_plot(; theme=:light) # hide - -text_color = theme == :light ? :black : :white # hide - -fig, ax, point_vec = do_setup(; theme = theme) # hide - -Y_zeros = zeros(length(Y_increments_geodesic)) -Y_geodesic_reshaped = [copy(Y_zeros), copy(Y_zeros), copy(Y_zeros)] -Y_cayley_reshaped = [copy(Y_zeros), copy(Y_zeros), copy(Y_zeros)] - -zip_ob = zip(Y_increments_geodesic, Y_increments_cayley, axes(Y_increments_geodesic, 1)) - -for (Y_increment_geodesic, Y_increment_cayley, i) in zip_ob - for d in (1, 2, 3) - Y_geodesic_reshaped[d][i] = Y_increment_geodesic[d] - - Y_cayley_reshaped[d][i] = Y_increment_cayley[d] - end -end - -scatter!(ax, Y_geodesic_reshaped...; - color = mred, label = rich("geodesic retraction"; color = text_color), markersize = 15) - -scatter!(ax, Y_cayley_reshaped...; - color = mblue, label = rich("Cayley retraction"; color = text_color), markersize = 15) - -arrow_vec = ([Δ[1]], [Δ[2]], [Δ[3]]) # hide -arrows!(ax, point_vec..., arrow_vec...; color = mred, linewidth = .02) # hide -backgroundcolor = theme == :light ? :white : :transparent -axislegend(; position = (.82, .75), backgroundcolor = backgroundcolor, color = text_color) # hide - -fig, ax, zip_ob, Y_increments_geodesic, Y_increments_cayley # hide -end # hide - -fig_light = make_plot(; theme = :light)[1] # hide -fig_dark = make_plot(; theme = :dark)[1] # hide - -CairoMakie.save("retraction_comparison_light.png", fig_light; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide -CairoMakie.save("retraction_comparison_dark.png", fig_dark; px_per_unit = Main.output_type == :html ? 1.5 : 2) # hide - -nothing -``` - -![Comparison between the geodesic and the Cayley retraction.](retraction_comparison_light.png) -![Comparison between the geodesic and the Cayley retraction.](retraction_comparison_dark.png) - -We see that for small ``\Delta`` increments the Cayley retraction seems to match the geodesic retraction very well, but for larger values there is a notable discrepancy. We can plot this discrepancy directly: - -```@setup s2_retraction -function plot_discrepancies(discrepancies; theme = :light) - fig = Figure(; backgroundcolor = :transparent) # hide - text_color = theme == :dark ? :white : :black # hide - ax = Axis(fig[1, 1]; # hide - backgroundcolor = :transparent, # hide - xlabel = rich("η", font = :italic, color = text_color), # hide - ylabel = rich("discrepancy", color = text_color), # hide - ) # hide - lines!(η_increments, discrepancies; label = rich("Discrepancies between geodesic and Cayley retraction", color = text_color), - linewidth = 2, color = mblue) - - axislegend(; position = (.22, .9), backgroundcolor = :transparent, color = text_color) # hide - - fig, ax -end -``` - -```@example s2_retraction -using LinearAlgebra: norm # hide -zip_ob = zip(Y_increments_geodesic, Y_increments_cayley, axes(Y_increments_geodesic, 1)) -_, __, zip_ob, Y_increments_geodesic, Y_increments_cayley = make_plot() # hide -discrepancies = [norm(Y_geo_inc - Y_cay_inc) for (Y_geo_inc, Y_cay_inc, _) in zip_ob] -fig_light = plot_discrepancies(discrepancies; theme = :light)[1] # hide -fig_dark = plot_discrepancies(discrepancies; theme = :dark)[1] # hide -CairoMakie.save("retraction_discrepancy_light.png", fig_light; px_per_unit = 1.3) # hide -CairoMakie.save("retraction_discrepancy_dark.png", fig_dark; px_per_unit = 1.3) # hide -nothing -``` - -![Discrepancy between the geodesic and the Cayley retraction.](retraction_discrepancy_light.png) -![Discrepancy between the geodesic and the Cayley retraction.](retraction_discrepancy_dark.png) - -## In `GeometricMachineLearning` - -The way we use *retractions*[^1] in `GeometricMachineLearning` is slightly different from their classical definition: - -[^1]: Classical retractions are also defined in `GeometricMachineLearning` under the same name, i.e. there is e.g. a method [`cayley(::StiefelLieAlgHorMatrix)`](@ref) and a method [`cayley(::StiefelManifold{T}, ::AbstractMatrix{T}) where T`](@ref) (the latter being the classical retraction); but the user is *strongly discouraged* from using classical retractions as these are computationally inefficient. - -```@eval -Main.definition(raw"Given a section ``\lambda:\mathcal{M}\to{}G,`` where ``\mathcal{M}`` is a homogeneous space, a **retraction** is a map ``\mathrm{Retraction}:\mathfrak{g}^\mathrm{hor}\to{}G`` such that -" * Main.indentation * raw"```math -" * Main.indentation * raw"\Delta \mapsto \lambda(Y)\mathrm{Retraction}(\lambda(Y)^{-1}\Omega(\Delta)\lambda(Y))E, -" * Main.indentation * raw"``` -" * Main.indentation * raw"is a classical retraction.") -``` - -This map ``\mathrm{Retraction}`` is also what was visualized in the figure on [the general optimization framework](@ref "Generalization to Homogeneous Spaces"). We now discuss how the geodesic retraction (exponential map) and the Cayley retraction are implemented in `GeometricMachineLearning`. - -## Retractions for Homogeneous Spaces - -Here we harness special properties of homogeneous spaces to obtain computationally efficient retractions for the [Stiefel manifold](@ref "The Stiefel Manifold") and the [Grassmann manifold](@ref "The Grassmann Manifold"). This is also discussed in e.g. [bendokat2020grassmann, bendokat2021real](@cite). - -The *geodesic retraction* is a retraction whose associated curve is also the unique geodesic. For many matrix Lie groups (including ``SO(N)``) geodesics are obtained by simply evaluating the exponential map [absil2008optimization, o1983semi](@cite): - -```@eval -Main.theorem(raw"The geodesic on a compact matrix Lie group ``G`` with bi-invariant metric for ``\bar{B}\in{}T_{\bar{A}}G`` is simply -" * Main.indentation * raw"```math -" * Main.indentation * raw"\gamma(t) = \exp(t\cdot{}\bar{B}\bar{A}^{-1})\bar{A} = A\exp(t\cdot{}\bar{A}^{-1}\bar{B}^n), -" * Main.indentation * raw"``` -" * Main.indentation * raw"where ``\exp:\mathfrak{g}\to{}G`` is the matrix exponential map.") -``` - -The last equality in the equation above is a result of: - -```math -\begin{aligned} -\exp(\bar{A}^{-1}\hat{B}\bar{A}) = \sum_{k=1}^\infty\frac{1}{k!}(\bar{A}^{-1}\hat{B}\bar{A})^k & = \sum_{k=1}^\infty \frac{1}{k!}\underbrace{(\bar{A}^{-1}\hat{B}\bar{A})\cdots(A^{-1}\hat{B}\bar{A})}_{\text{$k$ times}} \\ & = \sum_{k=1}^\infty \frac{1}{k!} \bar{A}^{-1} \hat{B}^k \bar{A} = \bar{A}^{-1}\exp(\hat{B})\bar{A}. -\end{aligned} -``` - -Because ``SO(N)`` is compact and we furnish it with the canonical metric, i.e. - -```math - g:T_{\bar{A}}G\times{}T_{\bar{A}}G \to \mathbb{R}, (B_1, B_2) \mapsto \mathrm{Tr}(B_1^TB_2) = \mathrm{Tr}((B_1\bar{A}^{-1})^T(B_2\bar{A}^{-1})), -``` - -its geodesics are thus equivalent to the exponential maps. We now use this observation to obtain an expression for the geodesics on the [Stiefel manifold](@ref "The Stiefel Manifold"). We use the following theorem from [o1983semi; Proposition 25.7](@cite): - -```@eval -Main.theorem(raw"The geodesics for a naturally reductive homogeneous space ``\mathcal{M}`` starting at ``Y`` are given by: -" * Main.indentation * raw"```math -" * Main.indentation * raw"\gamma_{\Delta}(t) = \exp(t\cdot\Omega(\Delta))Y, -" * Main.indentation * raw"``` -" * Main.indentation * raw"where the ``\exp`` is the exponential map for the Lie group ``G`` corresponding to ``\mathcal{M}``.") -``` - -The theorem requires the homogeneous space to be naturally reductive: - -```@eval -Main.definition(raw"A homogeneous space is called **naturally reductive** if the following two conditions hold: -" * Main.indentation * raw"1. ``\bar{A}^{-1}\bar{B}\bar{A}\in\mathfrak{g}^\mathrm{hor}`` for every ``\bar{B}\in\mathfrak{g}^\mathrm{hor}`` and ``\bar{A}\in\exp(\mathfrak{g}^\mathrm{ver}``), -" * Main.indentation * raw"2. ``g([X, Y]^\mathrm{hor}, Z) = g(X, [Y, Z]^\mathrm{hor})`` for all ``X, Y, Z \in \mathfrak{g}^\mathrm{hor}``, -" * Main.indentation * raw"where ``[X, Y]^\mathrm{hor} = \Omega(XYE - YXE)``. If only the first condition holds the homogeneous space is called **reductive** (but not **naturally reductive**).") -``` - -We state here without proof that the [Stiefel manifold](@ref "The Stiefel Manifold") and the [Grassmann manifold](@ref "The Grassmann Manifold") are naturally reductive. We can however provide empirical evidence here: - -```@example naturally_reductive -using GeometricMachineLearning # hide -import Random # hide -Random.seed!(123) # hide -B̄ = rand(SkewSymMatrix, 6) # ∈ 𝔤 -Ā = exp(B̄ - StiefelLieAlgHorMatrix(B̄, 3)) # ∈ exp(𝔤ᵛᵉʳ) - -X = rand(StiefelLieAlgHorMatrix, 6, 3) # ∈ 𝔤ʰᵒʳ -Y = rand(StiefelLieAlgHorMatrix, 6, 3) # ∈ 𝔤ʰᵒʳ -Z = rand(StiefelLieAlgHorMatrix, 6, 3) # ∈ 𝔤ʰᵒʳ - -@assert StiefelLieAlgHorMatrix(Ā' * X * Ā, 3) ≈ Ā' * X * Ā # hide -Ā' * X * Ā # this has to be in 𝔤ʰᵒʳ for St(3, 6) to be reductive -``` - -verifies the first property and - -```@example naturally_reductive -using LinearAlgebra: tr # hide -adʰᵒʳ(X, Y) = StiefelLieAlgHorMatrix(X * Y - Y * X, 3) - -@assert tr(adʰᵒʳ(X, Y)' * Z) ≈ tr(X' * adʰᵒʳ(Y, Z)) # hide -tr(adʰᵒʳ(X, Y)' * Z) ≈ tr(X' * adʰᵒʳ(Y, Z)) -``` - -verifies the second. - -In `GeometricMachineLearning` we always work with elements in ``\mathfrak{g}^\mathrm{hor}`` and the Lie group ``G`` is always ``SO(N)``. We hence use: - -```math - \gamma_\Delta(t) = \exp(\lambda(Y)\lambda(Y)^{-1}\Omega(\Delta)\lambda(Y)\lambda(Y)^{-1})Y = \lambda(Y)\exp(\lambda(Y)^{-1}\Omega(\Delta)\lambda(Y))E. -``` - -Based on this we define the maps: - -```math -\mathtt{geodesic}: \mathfrak{g}^\mathrm{hor} \to G, \bar{B} \mapsto \exp(\bar{B}), -``` - -and - -```math -\mathtt{cayley}: \mathfrak{g}^\mathrm{hor} \to G, \bar{B} \mapsto \mathrm{Cayley}(\bar{B}), -``` - -where ``\bar{B} = \lambda(Y)^{-1}\Omega(\Delta)\lambda(Y)``. These expressions for [`geodesic`](@ref geodesic(::StiefelLieAlgHorMatrix)) and [`cayley`](@ref cayley(::StiefelLieAlgHorMatrix)) are the ones that we typically use in `GeometricMachineLearning` for computational reasons. We show how we can utilize the sparse structure of ``\mathfrak{g}^\mathrm{hor}`` for computing the geodesic retraction and the Cayley retraction (i.e. the expressions ``\exp(\bar{B})`` and ``\mathrm{Cayley}(\bar{B})`` for ``\bar{B}\in\mathfrak{g}^\mathrm{hor}``). Similar derivations can be found in [celledoni2000approximating, fraikin2007optimization, bendokat2021real](@cite). - -```@eval -Main.remark(raw"Further note that, even though the global section ``\lambda:\mathcal{M} \to G`` is not unique, the final geodesic ``\gamma_\Delta(t) = \lambda(Y)\exp(\lambda(Y)^{-1}\Omega(\Delta)\lambda(Y))E`` does not depend on the particular section we choose.") -``` - -### The Geodesic Retraction - -An element ``\bar{B}`` of ``\mathfrak{g}^\mathrm{hor}`` can be written as: - -```math -\bar{B} = \begin{bmatrix} - A & -B^T \\ - B & \mathbb{O} -\end{bmatrix} = \begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix} =: B'(B'')^T, -``` - -where we exploit the sparse structure of the array, i.e. it is a multiplication of a ``N\times2n`` with a ``2n\times{}N`` matrix. - -We further use the following: - -```math - \begin{aligned} - \exp(B'(B'')^T) & = \sum_{n=0}^\infty \frac{1}{n!} (B'(B'')^T)^n = \mathbb{I} + \sum_{n=1}^\infty \frac{1}{n!} B'((B'')^TB')^{n-1}(B'')^T \\ - & = \mathbb{I} + B'\left( \sum_{n=1}^\infty \frac{1}{n!} ((B'')^TB')^{n-1} \right)B'' =: \mathbb{I} + B'\mathfrak{A}(B', B'')B'', - \end{aligned} -``` - -where we defined ``\mathfrak{A}(B', B'') := \sum_{n=1}^\infty \frac{1}{n!} ((B'')^TB')^{n-1}.`` Note that evaluating ``\mathfrak{A}`` relies on computing products of *small* matrices of size ``2n\times2n.`` We do this by relying on a simple Taylor expansion, implemented as `GeometricOptimizers.𝔄` (see the [`GeometricOptimizers` documentation](https://juliagni.github.io/GeometricOptimizers.jl/stable/) for its docstring). - -The final expression we obtain is: - -```math -\exp(\bar{B}) = \mathbb{I} + B' \mathfrak{A}(B', B'') (B'')^T -``` - -### The Cayley Retraction - -For the Cayley retraction we leverage the decomposition of ``\bar{B} = B'(B'')^T\in\mathfrak{g}^\mathrm{hor}`` through the *Sherman-Morrison-Woodbury formula*: - -```math -(\mathbb{I} - \frac{1}{2}B'(B'')^T)^{-1} = \mathbb{I} + \frac{1}{2}B'(\mathbb{I} - \frac{1}{2}B'(B'')^T)^{-1}(B'')^T -``` - -So what we have to compute the inverse of: - -```math -\mathbb{I} - \frac{1}{2}\begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix}\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} = -\begin{bmatrix} \mathbb{I} - \frac{1}{4}A & - \frac{1}{2}\mathbb{I} \\ \frac{1}{2}B^TB - \frac{1}{8}A^2 & \mathbb{I} - \frac{1}{4}A \end{bmatrix}. -``` - -By leveraging the sparse structure of the matrices in ``\mathfrak{g}^\mathrm{hor}`` we arrive at the following expression for the Cayley retraction (similar to the case of the geodesic retraction): - -```math -\mathrm{Cayley}(\bar{B}) = \mathbb{I} + \frac{1}{2} B' \left(\mathbb{I}_{2n} - \frac{1}{2} (B'')^T B'\right)^{-1} (B'')^T \left(\mathbb{I} + \frac{1}{2} \bar{B}\right), -``` - -where we have abbreviated ``\mathbb{I} := \mathbb{I}_N.`` We conclude with a remark: - -```@eval -Main.remark(raw"As mentioned previously the Lie group ``SO(N)``, i.e. the one corresponding to the Stiefel manifold and the Grassmann manifold, has a bi-invariant Riemannian metric associated with it: ``(B_1,B_2)\mapsto \mathrm{Tr}(B_1^TB_2)``. For other Lie groups (e.g. the symplectic group) the situation is slightly more difficult.") -``` - -One of such Lie groups is the *group of symplectic matrices* [bendokat2021real](@cite); for this group the expressions presented here are more complicated. - -## Library Functions - -The retraction framework itself lives in `GeometricOptimizers`; `GeometricMachineLearning` supplies -the manifold types and the retractions on them. - -```@docs -geodesic(::StiefelLieAlgHorMatrix) -geodesic(::GrassmannLieAlgHorMatrix) -cayley(::StiefelLieAlgHorMatrix) -cayley(::GrassmannLieAlgHorMatrix) -cayley(::StiefelManifold{T}, ::AbstractMatrix{T}) where T -``` - -```@raw latex -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -absil2008optimization -bendokat2021real -o1983semi -``` - -```@raw latex -\end{comment} -``` diff --git a/docs/src/optimizers/optimizer.md b/docs/src/optimizers/optimizer.md new file mode 100644 index 000000000..708b5b558 --- /dev/null +++ b/docs/src/optimizers/optimizer.md @@ -0,0 +1,29 @@ +# The `Optimizer` in `GeometricMachineLearning` + +The general framework for optimization on homogeneous spaces — the Riemannian gradient, the lift to +the global tangent space ``\mathfrak{g}^\mathrm{hor}``, the optimizer cache, the retraction and the +global section — belongs to `GeometricOptimizers` and is described in its documentation, under +[Optimization on Homogeneous Spaces](@extref GeometricOptimizers :doc:`manifold_optimizers`) and +[Retractions](@extref GeometricOptimizers :doc:`retractions`). + +What `GeometricMachineLearning` adds is the part that is about *neural networks*: walking the +parameter tree of a `NeuralNetwork`, applying the right update to each leaf — a retraction +for a weight on a manifold, ordinary arithmetic for a Euclidean one — and driving that from a data +loader over epochs and batches. + +The gradient comes from automatic differentiation on one batch at a time, so there is no objective +function to hand a line search; the step size is a property of the [`Optimizer`](@ref) instead. It is +either a number, or a `GeometricOptimizers.DecayingStatic` schedule: + +```julia +opt = Optimizer(Adam(Float32), nn; step_size = 1e-3) +opt = Optimizer(nn; AdamOptimizerWithDecay(n_epochs, Float32)...) +``` + +## Library Functions + +```@docs +Optimizer +optimize_for_one_epoch! +optimization_step! +``` diff --git a/docs/src/optimizers/optimizer_framework.md b/docs/src/optimizers/optimizer_framework.md deleted file mode 100644 index 7ae07d4c4..000000000 --- a/docs/src/optimizers/optimizer_framework.md +++ /dev/null @@ -1,116 +0,0 @@ -```@raw latex -In this chapter we introduce a \textit{general framework for manifold optimization} that is needed to efficiently train symplectic autoencoders. We start this chapter by discussing optimization for neural network in general and explain how we can generalize this to homogeneous spaces. We will see that an important ingredient for doing so are \textit{retractions} which we then elaborate on. After discussing how to make the computation of retractions efficient for homogeneous spaces we conclude the chapter by introducing the notion of \textit{parallel transport} which we need to extend the notion of \textit{momentum} in neural network optimization. -``` - -# Neural Network Optimizers - -In this section we present the general Optimizer framework used in `GeometricMachineLearning`. For more information on the particular steps involved in this consult the documentation on the various optimizer methods such as the gradient optimizer, the momentum optimizer and the [Adam optimizer](@ref "The Adam Optimizer"), and the documentation on [retractions](@ref "Retractions"). - -During *optimization* we aim at changing the neural network parameters in such a way to minimize the loss function. A loss function assigns a scalar value to the weights that [parametrize the neural network](@ref "Structure-Preserving Neural Networks"): - -```math - L: \mathbb{P}\to\mathbb{R}_{\geq0},\quad \Theta \mapsto L(\Theta), -``` - -where ``\mathbb{P}`` is the parameter space. We can then phrase the optimization task as: - -```@eval -Main.definition(raw"Given a neural network ``\mathcal{NN}`` parametrized by ``\Theta`` and a loss function ``L:\mathbb{P}\to\mathbb{R}`` we call an algorithm an **iterative optimizer** (or simply **optimizer**) if it performs the following task: -" * Main.indentation * raw"```math -" * Main.indentation * raw"\Theta \leftarrow \mathtt{Optimizer}(\Theta, \text{past history}, t), -" * Main.indentation * raw"``` -" * Main.indentation * raw"with the aim of decreasing the value ``L(\Theta)`` in each optimization step.") -``` - -The past history of the optimization is stored in the optimizer state managed -by `GeometricOptimizers`. GML's [`Optimizer`](@ref) combines that state with -neural-network parameters and the manifold retraction used for each update. - -Optimization for neural networks is (almost always) some variation on gradient descent. The most basic form of gradient descent is a discretization of the *gradient flow equation*: - -```math -\dot{\Theta} = -\nabla_\Theta{}L, -``` -by means of an Euler time-stepping scheme: -```math -\Theta^{t+1} = \Theta^{t} - h\nabla_{\Theta^{t}}L, -``` -where ``\eta`` (the time step of the Euler scheme) is referred to as the *learning rate*. - -This equation can easily be generalized to [manifolds](@ref "(Matrix) Manifolds") with the following two steps: -1. modify ``-\nabla_{\Theta^{t}}L\implies{}-h\mathrm{grad}_{\Theta^{t}}L,`` i.e. replace the Euclidean gradient by a [Riemannian gradient](@ref "The Riemannian Gradient") and -2. replace addition with the [geodesic map](@ref "Geodesic Sprays and the Exponential Map"). - -To sum up, we then have: - -```math -\Theta^{t+1} = \mathrm{geodesic}(\Theta^{t}, -h\mathrm{grad}_{\Theta^{t}}L). -``` - -In practice we very often do not use the geodesic map but approximations thereof. These approximations are called [retractions](@ref "Retractions"). - -## Generalization to Homogeneous Spaces - -In order to generalize neural network optimizers to [homogeneous spaces](@ref "Homogeneous Spaces") we utilize their corresponding [global tangent space representation](@ref "Global Tangent Spaces") ``\mathfrak{g}^\mathrm{hor}``. - -When introducing the notion of a [global tangent space](@ref "Global Tangent Spaces") we discussed how an element of the tangent space ``T_Y\mathcal{M}`` can be represented in ``\mathfrak{g}^\mathrm{hor}`` by performing two mappings: -1. the first one is the horizontal lift ``\Omega`` (see the docstring for [`GeometricMachineLearning.Ω`](@ref)) and -2. the second one is performing the adjoint operation[^1] of ``\lambda(Y),`` the section of ``Y``, on ``\Omega(\Delta).`` - -[^1]: By the *adjoint operation* ``\mathrm{ad}_A:\mathfrak{g}\to\mathfrak{g}`` for an element ``A\in{}G`` we mean ``B \mapsto A^{-1}BA``. - -The two steps together are performed as `global_rep` in `GeometricMachineLearning.` So we lift to ``\mathfrak{g}^\mathrm{hor}``: - -```math -\mathtt{global\_rep}: T_Y\mathcal{M} \to \mathfrak{g}^\mathrm{hor}, -``` - -and then perform all the steps of the optimizer in ``\mathfrak{g}^\mathrm{hor}.`` We can visualize all the steps required in the generalization of the optimizers: - -![Schematic visualization of neural network optimizers to homogeneous spaces.](../tikz/general_optimization_with_boundary_light.png) -![Schematic visualization of neural network optimizers to homogeneous spaces.](../tikz/general_optimization_with_boundary_dark.png) - -This picture summarizes all steps involved in an optimization step: -1. map the Euclidean gradient ``\nabla{}L\in\mathbb{R}^{N\times{}n}`` that was obtained via [automatic differentiation](@ref "Pullbacks and Automatic Differentiation") to the Riemannian gradient ``\mathrm{grad}L\in{}T_Y\mathcal{M}`` with the function [`rgrad`](@ref), -2. obtain the global tangent space representation of ``\mathrm{grad}L`` in ``\mathfrak{g}^\mathrm{hor}`` with the function `global_rep`, -3. perform an `update!`; this consists of two steps: (i) update the cache and (ii) output a *final velocity*, -4. use this final velocity to update the [global section](@ref "Global Sections") ``\Lambda\in{}G,`` -5. use the updated global section to update the neural network weight ``\in\mathcal{M}.`` This is done with `apply_section`. - -The `cache` stores information about previous optimization steps and is dependent on the optimizer. Typically the cache is represented as one or more elements in ``\mathfrak{g}^\mathrm{hor}``. Based on this the optimizer method (represented by `update!` in the figure) computes a *final velocity*. This final velocity is again an element of ``\mathfrak{g}^\mathrm{hor}``. The particular form of the cache and the updating rule depends on which [optimizer method we use](@ref "Standard Neural Network Optimizers"). - -The final velocity is then fed into a [retraction](@ref "Retractions")[^2]. For computational reasons we split the retraction into two steps, referred to as "Retraction" and `apply_section` above. These two mappings together are equivalent to: - -[^2]: A retraction is an approximation of the [geodesic map](@ref "Geodesic Sprays and the Exponential Map") - -```math -\mathrm{retraction}(\Delta) = \mathrm{retraction}(\lambda(Y)B^\Delta{}E) = \lambda(Y)\mathrm{Retraction}(B^\Delta), -``` - -where ``\Delta\in{}T_\mathcal{M}`` and ``B^\Delta`` is its representation in ``\mathfrak{g}^\mathrm{hor}`` as ``B^\Delta = \lambda(Y)^{-1}\Omega(\Delta)\lambda(Y).`` - - -## Library Functions - -```@docs -Optimizer -optimize_for_one_epoch! -optimization_step! -``` - -```@raw latex -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -brantner2023generalizing -``` - -```@raw latex -\end{comment} -``` diff --git a/docs/src/optimizers/optimizer_methods.md b/docs/src/optimizers/optimizer_methods.md deleted file mode 100644 index 71fb77c0e..000000000 --- a/docs/src/optimizers/optimizer_methods.md +++ /dev/null @@ -1,181 +0,0 @@ -```@raw latex -In the previous chapter we introduced a general optimizer framework without giving explicit examples of neural network optimizers; this is done here. This chapter discusses standard neural network optimizers, including gradient descent, momentum, and Adam. In the implementation of all these optimizers the \textit{optimizer cache} will play an important role. -``` - -# Standard Neural Network Optimizers - -In this section we discuss optimization methods that are often used in training neural networks. From a perspective of manifolds the *optimizer methods* outlined here operate on ``\mathfrak{g}^\mathrm{hor}`` only. Each of them has a cache associated with it[^1] and this cache is updated by `GeometricOptimizers.update!`. The precise role of this function is described below. - -[^1]: In the case of the [gradient optimizer](@ref "The Gradient Optimizer") this cache is trivial. - -## The Gradient Optimizer - -The gradient optimizer is the simplest optimization algorithm used to train neural networks. It was already briefly discussed when we introduced [Riemannian manifolds](@ref "Gradient Flows and Riemannian Optimization"). - -It simply does: - -```math -\mathrm{weight} \leftarrow \mathrm{weight} + (-\eta\cdot\mathrm{gradient}), -``` - -where addition has to be replaced with appropriate operations in the manifold case[^2]. - -[^2]: In the manifold case the expression ``-\eta\cdot\mathrm{gradient}`` is an element of the [global tangent space](@ref "Global Tangent Spaces") ``\mathfrak{g}^\mathrm{hor}`` and a retraction maps from ``\mathfrak{g}^\mathrm{hor}``. We then still have to compose it with the [updated global section](@ref "Parallel Transport") ``\Lambda^{(t)}``. - -The gradient method is constructed without a learning rate; pass the learning rate to [`Optimizer`](@ref) instead. - -```@example optimizer_methods -using GeometricMachineLearning # hide -import GeometricOptimizers # hide -const η = 0.01 -method = GradientMethod() -``` - -In order to use the optimizer we need an instance of [`Optimizer`](@ref) that is called with the method and the weights of the neural network: - - -```@example optimizer_methods -weight = (A = zeros(4, 4), ) -o = Optimizer(method, weight; step_size = η) - -nothing # hide -``` - -We then apply the optimizer to a derivative with [`optimization_step!`](@ref). This computes a *final velocity* from the cache, uses it to compute a retraction (or simply performs addition if we do not deal with a manifold) and writes the result back into the weights: - -```@example optimizer_methods -dx = (A = one(weight.A), ) -optimization_step!(o, GlobalSection(weight), weight, dx) - -weight.A -``` - -So what has happened here is that the gradient `dx` was simply multiplied with ``-\eta`` and added to the weight, as the cache of the gradient optimizer is trivial. - -## The Momentum Optimizer - -The momentum optimizer is similar to the gradient optimizer but further stores past information as *first moments*. We let these first moments *decay* with a *decay parameter* ``\alpha``: - -```math -\mathrm{weights} \leftarrow \mathrm{weights} + (\alpha\cdot\mathrm{moment} - \eta\cdot\mathrm{gradient}), -``` - -where addition has to be replaced with appropriate operations in the manifold case. - -In the case of the momentum optimizer the cache is non-trivial: - -```@example optimizer_methods -const α = 0.5 -method = MomentumMethod(α) -weight = (A = zeros(4, 4), ) -o = Optimizer(method, weight; step_size = η) - -# the moment is stored for each array in `weight` (which is a `NamedTuple`) -GeometricOptimizers.momentum(o.state).A -``` - -But as the moment is initialized with zeros it will lead to the same result as the gradient optimizer in the first iteration: - -```@example optimizer_methods -dx = (A = one(weight.A), ) -optimization_step!(o, GlobalSection(weight), weight, dx) - -weight.A -``` - -The cache has changed however: - -```@example optimizer_methods -GeometricOptimizers.momentum(o.state).A -``` - -If we have weights on manifolds calling [`Optimizer`](@ref) will automatically allocate the correct cache on ``\mathfrak{g}^\mathrm{hor}``: - -```@example optimizer_methods -weight = (Y = rand(StiefelManifold, 5, 3), ) - -GeometricOptimizers.momentum(Optimizer(method, weight).state).Y -``` - -So if the weight is ``Y\in{}St(n,N)`` the corresponding cache is initialized as the zero element on ``\mathfrak{g}^\mathrm{hor}\subset\mathbb{R}^{N\times{}N}`` as this is the global tangent space representation corresponding to the StiefelManifold. - -## The Adam Optimizer - -The Adam Optimizer is one of the most widely neural network optimizers. The cache of the Adam optimizer consists of *first and second moments*. The *first moments* ``B_1``, similar to the momentum optimizer, store linear information about the current and previous gradients, and the *second moments* ``B_2`` store quadratic information about current and previous gradients. These second moments can be interpreted as approximating the curvature of the optimization landscape. - -If all the weights are on a vector space, then we directly compute updates for ``B_1`` and ``B_2``: -1. ``B_1 \gets ((\rho_1 - \rho_1^t)/(1 - \rho_1^t))\cdot{}B_1 + (1 - \rho_1)/(1 - \rho_1^t)\cdot{}\nabla{}L,`` -2. ``B_2 \gets ((\rho_2 - \rho_1^t)/(1 - \rho_2^t))\cdot{}B_2 + (1 - \rho_2)/(1 - \rho_2^t)\cdot\nabla{}L\odot\nabla{}L,`` - -where ``\odot:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n`` is the *Hadamard product*: ``[a\odot{}b]_i = a_ib_i.`` ``\rho_1`` and ``\rho_2`` are hyperparameters. Their defaults, $\rho_1=0.9$ and $\rho_2=0.99$, are taken from [goodfellow2016deep; page 301](@cite). After having updated the `cache` (i.e. ``B_1`` and ``B_2``) we compute a *velocity* with which the parameters of the network are then updated: -* ``W_t\gets -\eta{}B_1/\sqrt{B_2 + \delta},`` -* ``Y^{(t+1)} \gets Y^{(t)} + W^{(t)},`` - -where the last addition has to be replaced with appropriate operations when dealing with manifolds. Further ``\eta`` is the *learning rate* and ``\delta`` is a small constant that is added for stability. The division, square root and addition in the computation of ``W_t`` are performed element-wise. - -In the following we show a schematic update that Adam performs for the case when no elements are on manifolds (also compare this figure with the [general optimization framework](@ref "Generalization to Homogeneous Spaces")): - -![Schematic representation of the Adam optimizer. The first Adam step updates the first and second moments, and the second Adam step outputs the final velocity.](../tikz/adam_optimizer_light.png) -![Schematic representation of the Adam optimizer. The first Adam step updates the first and second moments, and the second Adam step outputs the final velocity.](../tikz/adam_optimizer_dark.png) - -We demonstrate the Adam cache on the same example from before: -```@example optimizer_methods -const ρ₁ = 0.9 -const ρ₂ = 0.99 -const δ = 1e-8 - -method = Adam(Float64; β₁ = ρ₁, β₂ = ρ₂, δ) -o = Optimizer(method, weight; step_size = η) - -GeometricOptimizers.first_moment(o.state).Y -``` - -### Weights on Manifolds - -The problem with generalizing Adam to manifolds is that the Hadamard product ``\odot`` as well as the other element-wise operations (``/``, ``\sqrt{}`` and ``+``) lack a clear geometric interpretation. In `GeometricMachineLearning` we get around this issue by utilizing the [global tangent space representation](@ref "Global Tangent Spaces"). A similar approach is shown in [kong2023momentum](@cite). - -## The Adam Optimizer with Decay -The Adam optimizer with decay is similar to the standard Adam optimizer with the difference that the learning rate ``\eta`` decays exponentially. We start with a relatively high learning rate ``\eta_1`` (e.g. ``10^{-2}``) and end with a low learning rate ``\eta_2`` (e.g. ``10^{-8}``). If we want to use this optimizer we have to tell it beforehand how many epochs we train for such that it can adjust the learning rate decay accordingly: - -```@example optimizer_methods -const η₁ = 1e-2 -const η₂ = 1e-6 -const n_epochs = 1000 - -method = AdamOptimizerWithDecay(n_epochs, η₁, η₂, ρ₁, ρ₂, δ) -o = Optimizer(method, weight) - -nothing # hide -``` - - The cache is however exactly the same as for the Adam optimizer: - -```@example optimizer_methods -GeometricOptimizers.first_moment(o.state).Y -``` - -## Library Functions - -The method types and caches are provided by `GeometricOptimizers`; GML's -[`Optimizer`](@ref) adapts them to neural-network and manifold parameters. - -```@docs -AdamOptimizerWithDecay -``` - -```@raw latex -\begin{comment} -``` - -## References - -```@bibliography -Pages = [] -Canonical = false - -goodfellow2016deep -``` - -```@raw latex -\end{comment} -``` diff --git a/docs/src/outlook.md b/docs/src/outlook.md index a266e14e6..0132634ca 100644 --- a/docs/src/outlook.md +++ b/docs/src/outlook.md @@ -69,11 +69,11 @@ In this dissertation we applied the volume-preserving transformer for [learning ## Structure-Preserving Optimizers -Training a symplectic autoencoder requires optimization on manifolds[^4]. The particular manifolds we need in this case are "homogeneous spaces" [frankel2011geometry](@cite). In this dissertation we proposed a new optimizer framework that manages to [generalize existing neural network optimizers to manifolds](@ref "Neural Network Optimizers"). This is done by identifying a [global tangent space representation](@ref "Global Tangent Spaces") and dispenses with the need for a *projection step* as is necessary in other approaches [kong2023momentum, li2020efficient](@cite). +Training a symplectic autoencoder requires optimization on manifolds[^4]. The particular manifolds we need in this case are "homogeneous spaces" [frankel2011geometry](@cite). In this dissertation we proposed a new optimizer framework that manages to [generalize existing neural network optimizers to manifolds](@extref GeometricOptimizers The-optimizer-framework,-step-by-step). This is done by identifying a [global tangent space representation](@extref GeometricOptimizers Global-Tangent-Spaces) and dispenses with the need for a *projection step* as is necessary in other approaches [kong2023momentum, li2020efficient](@cite). [^4]: This is necessary to preserve the symplectic structure of the neural network. -As was already observed by others [zhang2021orthogonality, kong2023momentum, huang2018orthogonal](@cite) putting weights on manifolds can improve training significantly in contexts other than scientific computing. Motivated by this we show an example of training a vision transformer [dosovitskiy2020image](@cite) on the MNIST data set [deng2012mnist](@cite) to demonstrate the efficacy of the new optimizers. Contrary to other applications of the transformer we do not have to rely on layer normalization [xiong2020layer](@cite) or add connections to [achieve convergent training for relatively big neural networks](https://juliagni.github.io/GMLDatasets.jl/latest/mnist/mnist_tutorial/). We also applied the new optimizers to a neural network that contains weights on the [Grassmann manifold](@ref "The Grassmann Manifold") to be [able to sample from a nonlinear space](@ref "Example of a Neural Network with a Grassmann Layer"). +As was already observed by others [zhang2021orthogonality, kong2023momentum, huang2018orthogonal](@cite) putting weights on manifolds can improve training significantly in contexts other than scientific computing. Motivated by this we show an example of training a vision transformer [dosovitskiy2020image](@cite) on the MNIST data set [deng2012mnist](@cite) to demonstrate the efficacy of the new optimizers. Contrary to other applications of the transformer we do not have to rely on layer normalization [xiong2020layer](@cite) or add connections to [achieve convergent training for relatively big neural networks](https://juliagni.github.io/GMLDatasets.jl/latest/mnist/mnist_tutorial/). We also applied the new optimizers to a neural network that contains weights on the [Grassmann manifold](@extref GeometricOptimizers The-Grassmann-Manifold) to be [able to sample from a nonlinear space](@ref "Example of a Neural Network with a Grassmann Layer"). ## Outlook @@ -83,4 +83,4 @@ Symplectic autoencoders could be used for model reduction of higher-dimensional Structure-preserving transformers have shown great potential for learning dynamical systems, but their application should not be limited to that area. Structure-preserving machine learning techniques such as *Hamilton Monte Carlo* [duane1987hybrid](@cite) has been used in various fields such as image classification [cobb2021scaling](@cite) and inverse problems [fichtner2018hamiltonian](@cite) and we believe that the structure-preserving transformers introduced in this work can also find applications in these fields, by replacing the activation function in the attention layers of a vision transformer for example. -Lastly structure-preserving optimization is an exciting field, especially with regards to neural networks. The manifold optimizers introduced in this work can speed up neural network training significantly and are suitable for modern hardware (i.e. GPUs). They are however based on existing neural network optimizers such as Adam [kingma2014adam](@cite) and thus still lack a clear geometric interpretation. By utilizing a more geometric representation, as presented in this work, we hope to be able to find a differential equation describing Adam and other neural network optimizer, perhaps through a variational principle [wibisono2016variational, duruisseaux2022accelerated](@cite). One could also build on the existing optimization framework and use retractions other than the *geodesic retraction* and the *Cayley retraction* [presented here](@ref "Retractions"); an example would be a *QR-based retraction* [sato2019cholesky, gao2024optimization](@cite). This will be left for future work. \ No newline at end of file +Lastly structure-preserving optimization is an exciting field, especially with regards to neural networks. The manifold optimizers introduced in this work can speed up neural network training significantly and are suitable for modern hardware (i.e. GPUs). They are however based on existing neural network optimizers such as Adam [kingma2014adam](@cite) and thus still lack a clear geometric interpretation. By utilizing a more geometric representation, as presented in this work, we hope to be able to find a differential equation describing Adam and other neural network optimizer, perhaps through a variational principle [wibisono2016variational, duruisseaux2022accelerated](@cite). One could also build on the existing optimization framework and use retractions other than the *geodesic retraction* and the *Cayley retraction* [presented here](@extref GeometricOptimizers Retractions); an example would be a *QR-based retraction* [sato2019cholesky, gao2024optimization](@cite). This will be left for future work. \ No newline at end of file diff --git a/docs/src/pullbacks/computation_of_pullbacks.md b/docs/src/pullbacks/computation_of_pullbacks.md index 984e25416..4d48f5136 100644 --- a/docs/src/pullbacks/computation_of_pullbacks.md +++ b/docs/src/pullbacks/computation_of_pullbacks.md @@ -93,13 +93,13 @@ The notion of a *pullback in automatic differentiation* is motivated by the conc ```math f:\mathcal{V}\to\mathcal{W}, a \mapsto f(a) =: b, ``` -a *map of differentials* ``db \mapsto da``. In the differential geometry case ``db`` and ``da`` are part of the associated cotangent spaces, i.e. ``db\in{}T^*_b\mathcal{W}`` and ``da\in{}T^*_a\mathcal{V}``; in AD we (mostly) deal with spaces of arrays, i.e. vector spaces, which means that ``T^*_b\mathcal{W} \simeq \mathcal{W}`` and ``T^*_a\mathcal{V} \simeq \mathcal{V}``. If we have neural network weights on manifolds however, then we have to map weights from ``T^*_a\mathcal{V}`` (the result of an AD routine) to ``T_a\mathcal{V}`` before we can apply a [retraction](@ref "Retractions"). The mapping +a *map of differentials* ``db \mapsto da``. In the differential geometry case ``db`` and ``da`` are part of the associated cotangent spaces, i.e. ``db\in{}T^*_b\mathcal{W}`` and ``da\in{}T^*_a\mathcal{V}``; in AD we (mostly) deal with spaces of arrays, i.e. vector spaces, which means that ``T^*_b\mathcal{W} \simeq \mathcal{W}`` and ``T^*_a\mathcal{V} \simeq \mathcal{V}``. If we have neural network weights on manifolds however, then we have to map weights from ``T^*_a\mathcal{V}`` (the result of an AD routine) to ``T_a\mathcal{V}`` before we can apply a [retraction](@extref GeometricOptimizers Retractions). The mapping ```math T^*_a\mathcal{V} \to T_a\mathcal{V} ``` -is equivalent to applying the [Riemannian gradient](@ref "The Riemannian Gradient"). +is equivalent to applying the [Riemannian gradient](@extref GeometricOptimizers The-Riemannian-Gradient). ## Library Functions diff --git a/docs/src/reduced_order_modeling/losses.md b/docs/src/reduced_order_modeling/losses.md index c040be6d6..ba933a840 100644 --- a/docs/src/reduced_order_modeling/losses.md +++ b/docs/src/reduced_order_modeling/losses.md @@ -13,7 +13,7 @@ A popular trend in recent years has been considering known physical properties o ![The three ingredients that go into neural network-based machine learning.](../tikz/ingredients_light.png) ![The three ingredients that go into neural network-based machine learning.](../tikz/ingredients_dark.png) -Instead of considering certain properties through the loss function, we instead do so by enforcing them strongly through the network architecture and the optimizer; the latter pertains to [manifold optimization](@ref "Generalization to Homogeneous Spaces"). The advantages of this approach are the strong enforcement of properties that we know our network should have and much easier training because we do not have to tune hyperparameters. +Instead of considering certain properties through the loss function, we instead do so by enforcing them strongly through the network architecture and the optimizer; the latter pertains to [manifold optimization](@extref GeometricOptimizers Generalization-to-Homogeneous-Spaces). The advantages of this approach are the strong enforcement of properties that we know our network should have and much easier training because we do not have to tune hyperparameters. ## Projection and Reduction Errors of Reduced Models diff --git a/docs/src/reduced_order_modeling/pod_autoencoders.md b/docs/src/reduced_order_modeling/pod_autoencoders.md index aff7c3444..835fa42ae 100644 --- a/docs/src/reduced_order_modeling/pod_autoencoders.md +++ b/docs/src/reduced_order_modeling/pod_autoencoders.md @@ -49,7 +49,7 @@ Main.definition(raw"An **autoencoder** is a tuple of two mappings ``(\mathcal{P} " * Main.indentation * raw"During training we optimize the autoencoder for minimizing the *projection error*.") ``` -Unlike in the POD case we have to resort to using [neural network optimizers](@ref "Neural Network Optimizers") in order to adapt the neural network to the data at hand as opposed to simply using SVD. The use of autoencoders instead of POD is extremely advantageous in the case when we deal with problems that exhibit a slowly-decaying Kolmogorov ``n``-width. During training we minimize the [projection error](@ref "Projection Error"). +Unlike in the POD case we have to resort to using [neural network optimizers](@extref GeometricOptimizers The-optimizer-framework,-step-by-step) in order to adapt the neural network to the data at hand as opposed to simply using SVD. The use of autoencoders instead of POD is extremely advantageous in the case when we deal with problems that exhibit a slowly-decaying Kolmogorov ``n``-width. During training we minimize the [projection error](@ref "Projection Error"). ```@eval Main.remark(raw"Note that POD can be seen as a special case of an autoencoder where the encoder and the decoder both consist of only one matrix. If we restrict this matrix to be orthonormal, i.e. optimize on the Stiefel manifold, then the best solution we can obtain is equivalent to applying SVD and finding the POD basis.") diff --git a/docs/src/reduced_order_modeling/symplectic_mor.md b/docs/src/reduced_order_modeling/symplectic_mor.md index bb71c8412..a6d1499a0 100644 --- a/docs/src/reduced_order_modeling/symplectic_mor.md +++ b/docs/src/reduced_order_modeling/symplectic_mor.md @@ -49,7 +49,7 @@ Main.theorem(raw"A Hamiltonian system on the reduced space ``(\mathbb{R}^{2n}, \ " * Main.indentation * raw"so the dynamics on ``\mathcal{M}`` can be described through a Hamiltonian ODE on ``\mathbb{R}^{2n}.``") ``` -For the proof we use the fact that ``\mathcal{M} = \mathcal{R}(\mathbb{R}^{2n})`` is a manifold [whose coordinate chart is the local inverse](@ref "The Immersion Theorem") of ``\mathcal{R}`` which we will call ``\psi``, i.e. around a point ``y\in\mathcal{M}`` we have ``\psi\circ\mathcal{R}(y) = y.``[^3] We further define the *symplectic inverse* of a matrix ``A\in\mathbb{R}^{2N\times2n}`` as +For the proof we use the fact that ``\mathcal{M} = \mathcal{R}(\mathbb{R}^{2n})`` is a manifold [whose coordinate chart is the local inverse](@extref GeometricOptimizers The-Immersion-Theorem) of ``\mathcal{R}`` which we will call ``\psi``, i.e. around a point ``y\in\mathcal{M}`` we have ``\psi\circ\mathcal{R}(y) = y.``[^3] We further define the *symplectic inverse* of a matrix ``A\in\mathbb{R}^{2N\times2n}`` as [^3]: A similar proof can be found in [yildiz2024data](@cite). Further note that, if we enforced the condition ``\mathcal{P}\circ\mathcal{R} = \mathrm{id}`` exactly, the projection ``\mathcal{P}`` would be equal to the local inverse ``\psi.`` For the proof here we however only require the existence of ``\psi``, not its explicit construction as ``\mathcal{P}.`` @@ -104,7 +104,7 @@ For proper symplectic decomposition (PSD) the reduction ``\mathcal{P}`` and the ```math \mathcal{R} \equiv \Psi_\mathrm{CL} = \begin{bmatrix} \Phi & \mathbb{O} \\ \mathbb{O} & \Phi \end{bmatrix} \text{ where $\Phi\in{}St(n,N)\subset\mathbb{R}^{N\times{}n}$}, ``` -i.e. both ``\Phi`` and ``\Psi_\mathrm{CL}`` are elements of the [Stiefel manifold](@ref "The Stiefel Manifold") and we furthermore have ``\Psi_\mathrm{CL}^T\mathbb{J}_{2N}\Psi_\mathrm{CL} = \mathbb{J}_{2n}``, i.e. ``\Psi_\mathrm{CL}`` is symplectic. If the [snapshot matrix](@ref "Snapshot Matrix") is of the form: +i.e. both ``\Phi`` and ``\Psi_\mathrm{CL}`` are elements of the [Stiefel manifold](@extref GeometricOptimizers The-Stiefel-Manifold) and we furthermore have ``\Psi_\mathrm{CL}^T\mathbb{J}_{2N}\Psi_\mathrm{CL} = \mathbb{J}_{2n}``, i.e. ``\Psi_\mathrm{CL}`` is symplectic. If the [snapshot matrix](@ref "Snapshot Matrix") is of the form: ```math M = \left[\begin{array}{c:c:c:c} diff --git a/docs/src/structure_preservation/structure_preserving_neural_networks.md b/docs/src/structure_preservation/structure_preserving_neural_networks.md index 2a445aed9..bfa6daa84 100644 --- a/docs/src/structure_preservation/structure_preserving_neural_networks.md +++ b/docs/src/structure_preservation/structure_preserving_neural_networks.md @@ -10,7 +10,7 @@ Main.definition(raw"A **neural network architecture** is a parameter-dependent r " * Main.indentation * raw"where ``\Theta`` are the *parameters of the neural network* (we call ``\mathbb{P}`` the parameter space). ``\mathbb{P}``, the domain space ``\mathcal{D}`` and the target space ``\mathcal{M}`` of the neural network may be spaces with arbitrary structure in general (i.e. need not be vector spaces).") ``` -In this text the spaces ``\mathcal{D}`` and ``\mathcal{M}`` are vector spaces in most cases[^1]. The parameter space ``\mathbb{P}`` is however build [from manifolds in many cases](@ref "Neural Network Optimizers"). Weights have to be put on manifolds to realize [certain architectures that would otherwise not be possible](@ref "The Symplectic Autoencoder") and can make training [more efficient in other cases](https://juliagni.github.io/GMLDatasets.jl/latest/mnist/mnist_tutorial/). +In this text the spaces ``\mathcal{D}`` and ``\mathcal{M}`` are vector spaces in most cases[^1]. The parameter space ``\mathbb{P}`` is however build [from manifolds in many cases](@extref GeometricOptimizers The-optimizer-framework,-step-by-step). Weights have to be put on manifolds to realize [certain architectures that would otherwise not be possible](@ref "The Symplectic Autoencoder") and can make training [more efficient in other cases](https://juliagni.github.io/GMLDatasets.jl/latest/mnist/mnist_tutorial/). [^1]: One exception is [Grassmann learning](@ref "Example of a Neural Network with a Grassmann Layer") where we learn a vector space. diff --git a/docs/src/structure_preservation/symplecticity.md b/docs/src/structure_preservation/symplecticity.md index 68845e10b..1e46a0eaa 100644 --- a/docs/src/structure_preservation/symplecticity.md +++ b/docs/src/structure_preservation/symplecticity.md @@ -17,7 +17,7 @@ Main.definition(raw"A **symplectic structure** or **symplectic 2-form** ``\Omega We forego the precise definition of *closedness* because it would require us to introduce differential forms [arnold1978mathematical, bishop1980tensor](@cite). This property is also closely related to the *Jacobi identity* [kraus2017gempic; Chapter 4.4](@cite). After having defined a symplectic structure, we can introduce *Hamiltonian vector fields*[^1]: -[^1]: Also compare this to the definition of the [Riemannian gradient](@ref "The Riemannian Gradient"). +[^1]: Also compare this to the definition of the [Riemannian gradient](@extref GeometricOptimizers The-Riemannian-Gradient). ```@eval Main.definition(raw"A **Hamiltonian vector field** at ``x\in\mathcal{M}`` corresponding to the function ``H:\mathcal{M}\to\mathbb{R}`` (called **the Hamiltonian**) is a vector field that has the following property: @@ -59,7 +59,7 @@ We can then reformulate a Hamiltonian vector field as two separate vector fields ## Solution of Symplectic Systems -The [flow](@ref "The Existence-And-Uniqueness Theorem") of a Hamiltonian ODE has very restrictive properties, the most important one of these is called *symplecticity* [hairer2006geometric](@cite). This property dramatically restricts the dynamically accessible states of the flow map. For a canonical Hamiltonian system symplecticity is defined as follows: +The [flow](@extref GeometricOptimizers The-Existence-And-Uniqueness-Theorem) of a Hamiltonian ODE has very restrictive properties, the most important one of these is called *symplecticity* [hairer2006geometric](@cite). This property dramatically restricts the dynamically accessible states of the flow map. For a canonical Hamiltonian system symplecticity is defined as follows: ```@eval diff --git a/docs/src/tutorials/grassmann_layer.md b/docs/src/tutorials/grassmann_layer.md index 1f2e6ee4c..5bbb90a7a 100644 --- a/docs/src/tutorials/grassmann_layer.md +++ b/docs/src/tutorials/grassmann_layer.md @@ -4,7 +4,7 @@ In this chapter we give another example of using the new neural network optimize # Example of a Neural Network with a Grassmann Layer -Here we show how to implement a neural network that contains a layer whose weight is an element of the [Grassmann manifold](@ref "The Grassmann Manifold") and where this is useful. Recall that the Grassmann manifold ``Gr(n, N)`` is the set of vector spaces of dimension ``n`` embedded in ``\mathbb{R}^N``. So if we optimize on the Grassmann manifold, we optimize for an *ideal* ``n``-dimensional vector space in the bigger space ``\mathbb{R}^N``. +Here we show how to implement a neural network that contains a layer whose weight is an element of the [Grassmann manifold](@extref GeometricOptimizers The-Grassmann-Manifold) and where this is useful. Recall that the Grassmann manifold ``Gr(n, N)`` is the set of vector spaces of dimension ``n`` embedded in ``\mathbb{R}^N``. So if we optimize on the Grassmann manifold, we optimize for an *ideal* ``n``-dimensional vector space in the bigger space ``\mathbb{R}^N``. We visualize this: diff --git a/docs/src/tutorials/linear_symplectic_transformer.md b/docs/src/tutorials/linear_symplectic_transformer.md index e3692cec2..347200d90 100644 --- a/docs/src/tutorials/linear_symplectic_transformer.md +++ b/docs/src/tutorials/linear_symplectic_transformer.md @@ -63,11 +63,11 @@ nn_standard = NeuralNetwork(arch_standard) nn_symplectic = NeuralNetwork(arch_symplectic) nn_sympnet = NeuralNetwork(arch_sympnet) -o_method = AdamOptimizerWithDecay(n_epochs; T = Float64) +o_pairing = AdamOptimizerWithDecay(n_epochs, Float64) -o_standard = Optimizer(o_method, nn_standard) -o_symplectic = Optimizer(o_method, nn_symplectic) -o_sympnet = Optimizer(o_method, nn_sympnet) +o_standard = Optimizer(nn_standard; o_pairing...) +o_symplectic = Optimizer(nn_symplectic; o_pairing...) +o_sympnet = Optimizer(nn_sympnet; o_pairing...) batch = Batch(batch_size, seq_length) batch2 = Batch(batch_size) diff --git a/docs/src/tutorials/symplectic_autoencoder.md b/docs/src/tutorials/symplectic_autoencoder.md index e88db5f1e..5bcb11e7c 100644 --- a/docs/src/tutorials/symplectic_autoencoder.md +++ b/docs/src/tutorials/symplectic_autoencoder.md @@ -119,7 +119,7 @@ psd_nn_cpu = NeuralNetwork(psd_arch, CPU(), eltype(dl_cpu)) solve!(psd_nn_cpu, dl_cpu) ``` -The `SymplecticAutoencoder` we train with [`AdamOptimizerWithDecay`](@ref) however[^2]: +The `SymplecticAutoencoder` we train with [`AdamOptimizerWithDecay`](@extref GeometricOptimizers The-Adam-Optimizer-with-Decay) however[^2]: [^2]: It is not feasible to perform the training on CPU, which is why we use `CUDA` [besard2018juliagpu](@cite) here. We further perform the training in single precision. @@ -134,7 +134,7 @@ dl = DataLoader(dl_cpu, backend, Float32) sae_nn_gpu = NeuralNetwork(sae_arch, CUDADevice(), Float32) -o = Optimizer(AdamOptimizerWithDecay(integrator_train_epochs), sae_nn_gpu) +o = Optimizer(sae_nn_gpu; AdamOptimizerWithDecay(integrator_train_epochs)...) # train the network o(sae_nn_gpu, dl, Batch(batch_size), n_epochs) @@ -269,9 +269,9 @@ integrator_architecture = StandardTransformerIntegrator(reduced_dim; integrator_nn = NeuralNetwork(integrator_architecture, backend) -integrator_method = AdamOptimizerWithDecay(integrator_train_epochs) +integrator_pairing = AdamOptimizerWithDecay(integrator_train_epochs) -o_integrator = Optimizer(integrator_method, integrator_nn) +o_integrator = Optimizer(integrator_nn; integrator_pairing...) dl = dl_cpu # hide # map from autoencoder type to integrator type @@ -464,8 +464,8 @@ const integrator_architecture2 = StandardTransformerIntegrator(reduced_dim2; L = 3, upscaling_activation = tanh) integrator_nn2 = NeuralNetwork(integrator_architecture2, backend) -const integrator_method2 = AdamOptimizerWithDecay(integrator_train_epochs) -const o_integrator2 = Optimizer(integrator_method2, integrator_nn2) +const integrator_pairing2 = AdamOptimizerWithDecay(integrator_train_epochs) +const o_integrator2 = Optimizer(integrator_nn2; integrator_pairing2...) loss2 = GeometricMachineLearning.ReducedLoss(encoder(psd_nn2), decoder(psd_nn2)) nothing # hide diff --git a/docs/src/tutorials/symplectic_transformer.md b/docs/src/tutorials/symplectic_transformer.md index c5d598b13..1dc1e7f0e 100644 --- a/docs/src/tutorials/symplectic_transformer.md +++ b/docs/src/tutorials/symplectic_transformer.md @@ -63,11 +63,11 @@ nn_standard = NeuralNetwork(arch_standard) nn_symplectic = NeuralNetwork(arch_symplectic) nn_sympnet = NeuralNetwork(arch_sympnet) -o_method = AdamOptimizerWithDecay(n_epochs; T = Float64) +o_pairing = AdamOptimizerWithDecay(n_epochs, Float64) -o_standard = Optimizer(o_method, nn_standard) -o_symplectic = Optimizer(o_method, nn_symplectic) -o_sympnet = Optimizer(o_method, nn_sympnet) +o_standard = Optimizer(nn_standard; o_pairing...) +o_symplectic = Optimizer(nn_symplectic; o_pairing...) +o_sympnet = Optimizer(nn_sympnet; o_pairing...) batch = Batch(batch_size, seq_length) batch2 = Batch(batch_size) diff --git a/docs/src/tutorials/sympnet_tutorial.md b/docs/src/tutorials/sympnet_tutorial.md index a1c8643b0..602f58a2c 100644 --- a/docs/src/tutorials/sympnet_tutorial.md +++ b/docs/src/tutorials/sympnet_tutorial.md @@ -96,7 +96,7 @@ parameterlength(g_nn.model) Main.remark(raw"We can also specify whether we would like to start with a layer that changes the ``q``-component or one that changes the ``p``-component. This can be done via the keywords `init_upper` for the `GSympNet`, and `init_upper_linear` and `init_upper_act` for the `LASympNet`.") ``` -We have to define an [optimizer](@ref "Standard Neural Network Optimizers") which will be used in training of the SympNet. In this example we use [Adam](@ref "The Adam Optimizer"): +We have to define an [optimizer](@extref GeometricOptimizers Standard-Neural-Network-Optimizers) which will be used in training of the SympNet. In this example we use [Adam](@extref GeometricOptimizers The-Adam-Optimizer): ```@example sympnet # set up optimizer; for this we first need to specify the optimization method diff --git a/docs/src/tutorials/volume_preserving_transformer_rigid_body.md b/docs/src/tutorials/volume_preserving_transformer_rigid_body.md index c7ca10952..8f929ed21 100644 --- a/docs/src/tutorials/volume_preserving_transformer_rigid_body.md +++ b/docs/src/tutorials/volume_preserving_transformer_rigid_body.md @@ -183,18 +183,18 @@ nn_st = NeuralNetwork(arch_st, backend, T) (parameterlength(nn_vpff), parameterlength(nn_vpt), parameterlength(nn_st)) ``` -We now train the various networks. For this we use [`AdamOptimizerWithDecay`](@ref): +We now train the various networks. For this we use [`AdamOptimizerWithDecay`](@extref GeometricOptimizers The-Adam-Optimizer-with-Decay): ```@example rigid_body const n_epochs = 500000 const batch_size = 16384 const feedforward_batch = Batch(batch_size) const transformer_batch = Batch(batch_size, seq_length, seq_length) -const opt_method = AdamOptimizerWithDecay(n_epochs, 1e-2, 1e-6; T = T) +const opt_pairing = AdamOptimizerWithDecay(n_epochs, T; η₁ = 1e-2, η₂ = 1e-6) -o_vpff = Optimizer(opt_method, nn_vpff) -o_vpt = Optimizer(opt_method, nn_vpt) -o_st = Optimizer(opt_method, nn_st) +o_vpff = Optimizer(nn_vpff; opt_pairing...) +o_vpt = Optimizer(nn_vpt; opt_pairing...) +o_st = Optimizer(nn_st; opt_pairing...) nothing # hide ``` ```julia From ed30aee8ab0fbb3a269f89d5e8e80256d0521eba Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Mon, 17 Aug 2026 21:58:58 +0900 Subject: [PATCH 04/12] Bring the scripts to the new AdamOptimizerWithDecay MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit `AdamOptimizerWithDecay` returns an `(algorithm, linesearch)` pairing now rather than an `OptimizerMethod`, so the three scripts that used it splat it into `Optimizer` instead of passing it positionally. The bindings are renamed from `…_method` to `…_pairing` to say which of the two it is -- a pairing passed where a method is expected is otherwise a `MethodError` several frames away from the line that caused it. `scripts/Project.toml` gains `[sources] GeometricMachineLearning = {path = ".."}`, replacing a comment that said the path should stay out of the file "so that it works from any clone". That reasoning predates `[sources]`: the path is relative to the file, so it does work from any clone, and it removes the one-off `Pkg.develop(path = "..")` the comment asked for. This edit was in the working tree before the branch and is kept deliberately. Co-Authored-By: Claude Opus 5 (1M context) --- scripts/Project.toml | 5 +- scripts/enzyme.jl | 39 ++++ scripts/harmonic_oscillator_sympnet.jl | 58 ++++++ scripts/sae_script2.jl | 181 ++++++++++++++++++ .../symplectic_autoencoders/online_sympnet.jl | 8 +- .../online_transformer_for_sae.jl | 4 +- .../rigid_body.jl | 4 +- scripts/zygote.jl | 20 ++ 8 files changed, 308 insertions(+), 11 deletions(-) create mode 100644 scripts/enzyme.jl create mode 100644 scripts/harmonic_oscillator_sympnet.jl create mode 100644 scripts/sae_script2.jl create mode 100644 scripts/zygote.jl diff --git a/scripts/Project.toml b/scripts/Project.toml index 6443112b5..8e88c3648 100644 --- a/scripts/Project.toml +++ b/scripts/Project.toml @@ -28,6 +28,5 @@ Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7" Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" -# These scripts run against a development checkout of GeometricMachineLearning. Activate this -# environment and `Pkg.develop(path = "..")` once; the path stays out of this file so that it works -# from any clone. +[sources] +GeometricMachineLearning = {path = ".."} diff --git a/scripts/enzyme.jl b/scripts/enzyme.jl new file mode 100644 index 000000000..78612cd4a --- /dev/null +++ b/scripts/enzyme.jl @@ -0,0 +1,39 @@ +using BenchmarkTools +using Enzyme +using LinearAlgebra +using Zygote + + +loss(A,x) = norm(A*x) + +# function loss(A,x) +# y = zero(x) +# mul!(y,A,x) +# norm(y) +# end + +function test(n) + A = rand(n,n) + x = rand(n) + + l = a -> loss(a,x) + + dA = zero(A) + + println("\nn = $n") + + println("\nEnzyme (autodiff):") + @btime Enzyme.autodiff(Reverse, $l, Active, Duplicated($A, $dA)) + + println("\nEnzyme (gradient):") + @btime Enzyme.gradient(Reverse, $l, $A) + + println("\nZygote:") + @btime Zygote.gradient($l, $A)[1] + + println("") +end + +test(100) +test(1000) +test(10000) diff --git a/scripts/harmonic_oscillator_sympnet.jl b/scripts/harmonic_oscillator_sympnet.jl new file mode 100644 index 000000000..4f48ec5d6 --- /dev/null +++ b/scripts/harmonic_oscillator_sympnet.jl @@ -0,0 +1,58 @@ +using GeometricMachineLearning +using GeometricIntegrators: ImplicitMidpoint, integrate +import GeometricProblems.HarmonicOscillator as ho + +# the problem is the ODE of the harmonic oscillator +ho_problem = ho.hodeproblem(; tspan = 500) + +# integrate the system +solution = integrate(ho_problem, ImplicitMidpoint()) + +dl_raw = DataLoader(solution; suppress_info = true) + +# specify the data type and the backend +type = Float64 +backend = CPU() + +# we can then make a new instance of `DataLoader` with this backend and type. +dl = DataLoader(dl_raw, backend, type) + + +const upscaling_dimension = 2 +const nhidden = 1 +const activation = tanh +const n_layers = 4 # number of layers for the G-SympNet +const depth = 4 # number of layers in each linear block in the LA-SympNet + +# calling G-SympNet architecture +gsympnet = GSympNet(dl; upscaling_dimension = upscaling_dimension, + n_layers = n_layers, + activation = activation) + +# initialize the networks +g_nn = NeuralNetwork(gsympnet, backend, type) + +# set up optimizer; for this we first need to specify the optimization method +opt_method = AdamOptimizer(type) + +# we then call the optimizer struct which allocates the cache +g_opt = Optimizer(opt_method, g_nn) + +# determine the batch size (the number of samples in one batch) +const batch_size = 16 + +batch = Batch(batch_size) + +# number of training epochs +const nepochs = 100 + +# perform training (returns array that contains the total loss for each training step) +g_loss_array = g_opt(g_nn, dl, batch, nepochs; show_progress = false) + +ics = (q=dl.input.q[:, 1, 1], p=dl.input.p[:, 1, 1]) + +steps_to_plot = 1000 + +#predictions +g_trajectory = iterate(g_nn, ics; n_points = steps_to_plot) + diff --git a/scripts/sae_script2.jl b/scripts/sae_script2.jl new file mode 100644 index 000000000..902a8db61 --- /dev/null +++ b/scripts/sae_script2.jl @@ -0,0 +1,181 @@ +using GeometricIntegrators: integrate, ImplicitMidpoint +using GeometricMachineLearning +import Random # hide +import GeometricProblems.TodaLattice as tl +using JLD2 +using CairoMakie + +sae_dir = "animations" +mkpath(sae_dir) + +N = tl.Ñ # hide +Δx = 1. / (N - 1) # hide +Ω = -0.5 : Δx : 0.5 # hide +tl.μ + +# todo +#pr = tl.hodeproblem(; tspan = (0.0, 8.)) +pr = tl.hodeproblem(; tspan = (0.0, 800.)) +@time "FOM + Implicit Midpoint" sol = integrate(pr, ImplicitMidpoint()) + +dl_cpu = DataLoader(sol; autoencoder = true, suppress_info = true) + +const reduced_dim = 2 + +Random.seed!(123) # hide +sae_arch = SymplecticAutoencoder(dl_cpu.input_dim, reduced_dim; n_encoder_blocks = 4, + n_decoder_blocks = 4, + n_encoder_layers = 2, + n_decoder_layers = 2) + +const mtc = GeometricMachineLearning.map_to_cpu + +sae_trained_parameters = load("../docs/src/tutorials/sae_parameters.jld2")["sae_parameters"] +_nnp(ps::Tuple) = NeuralNetworkParameters{Tuple(Symbol("L$(i)") for i in 1:length(ps))}(ps) +sae_nn_cpu = NeuralNetwork(sae_arch, Chain(sae_arch), _nnp(sae_trained_parameters), CPU()) + +sae_rs = HRedSys(pr, encoder(sae_nn_cpu), decoder(sae_nn_cpu); integrator = ImplicitMidpoint()) + +# @time "FOM + Implicit Midpoint" sol_full = integrate_full_system(sae_rs) # hide +@time "SAE + Implicit Midpoint" sol_sae_reduced = integrate_reduced_system(sae_rs) # hide + + + + + +const T = Float32 +_T(qp::NamedTuple{(:q, :p)}) = (q = T.(qp.q), p = T.(qp.p)) + +dl_reduced = DataLoader(encoder(sae_nn_cpu)(_T(dl_cpu.input))) + +# lines(dl_reduced.input.q[1, :, 1], dl_reduced.input.p[1, :, 1]) + +# sympnet_arch = GSympNet(2; n_layers = 10) +# sympnet_nn = NeuralNetwork(sympnet_arch, T) +# o = Optimizer(AdamOptimizer(), sympnet_nn) +# o(sympnet_nn, dl_reduced, Batch(10), 500) + +morange = RGBf(255 / 256, 127 / 256, 14 / 256) +mred = RGBf(214 / 256, 39 / 256, 40 / 256) +mpurple = RGBf(148 / 256, 103 / 256, 189 / 256) +mblue = RGBf(31 / 256, 119 / 256, 180 / 256) +mgreen = RGBf(44 / 256, 160 / 256, 44 / 256) + +function plot_solution(time_step; theme = :light, framerate = 50) + textcolor = theme == :dark ? :white : :black + fig = Figure(size = (1000, 500), figure_padding = (5,50,5,10), fontsize = 24) + ax = Axis(fig[1, 1], backgroundcolor = :transparent, + bottomspinecolor = textcolor, + topspinecolor = textcolor, + leftspinecolor = textcolor, + rightspinecolor = textcolor, + xtickcolor = textcolor, + ytickcolor = textcolor, + xticklabelcolor = textcolor, + yticklabelcolor = textcolor, + xlabel=L"\omega", + ylabel=L"q", + xlabelcolor = textcolor, + ylabelcolor = textcolor) + lines!(ax, sol.s.q[time_step], label = rich("FOM + Implicit Midpoint"; color = textcolor), color = mblue) + lines!(ax, sae_rs.decoder((q = sol_sae_reduced.s.q[time_step], p = sol_sae_reduced.s.p[time_step])).q, label = rich("SAE + Implicit Midpoint"; color = textcolor), color = mgreen) + axislegend(ax; position = :rt) + xlims!(ax, 0, 200) + ylims!(ax, 0, 1) + fig +end + + +#### Transformer + +const seq_length = 4 +integrator_architecture = StandardTransformerIntegrator(reduced_dim; + transformer_dim = 20, + n_blocks = 3, + n_heads = 5, + L = 3, + upscaling_activation = tanh) + +nn_integrator_parameters = load("../docs/src/tutorials/integrator_parameters.jld2")["integrator_parameters"] # hide +integrator_nn = NeuralNetwork(integrator_architecture, Chain(integrator_architecture), _nnp(nn_integrator_parameters), CPU()) # hide + +# todo +#n_time_steps = 100 +n_time_steps = 10000 + +ics = (q = dl_reduced.input.q[:, 1:seq_length], p = dl_reduced.input.p[:, 1:seq_length]) +time_series = iterate(mtc(integrator_nn), ics; n_points = n_time_steps, prediction_window = seq_length) +function plot_solution2(time_step; theme = :light, framerate = 50) + textcolor = theme == :dark ? :white : :black + fig = Figure(size = (1000, 500), figure_padding = (5,50,5,10), fontsize = 24) + ax = Axis(fig[1, 1], backgroundcolor = :transparent, + bottomspinecolor = textcolor, + topspinecolor = textcolor, + leftspinecolor = textcolor, + rightspinecolor = textcolor, + xtickcolor = textcolor, + ytickcolor = textcolor, + xticklabelcolor = textcolor, + yticklabelcolor = textcolor, + xlabel=L"\omega", + ylabel=L"q", + xlabelcolor = textcolor, + ylabelcolor = textcolor) + lines!(ax, sol.s.q[time_step], label = rich("FOM + Implicit Midpoint"; color = textcolor), color = mblue) + # prediction = (q = time_series.q[:, end], p = time_series.p[:, end]) + prediction = (q = time_series.q[:, time_step], p = time_series.p[:, time_step]) + prediction_big = decoder(sae_nn_cpu)(prediction) + + lines!(ax, prediction_big.q; label = rich("SAE + Transformer"; color = textcolor), color = mpurple) + axislegend(ax; position = :rt) + xlims!(ax, 0, 200) + ylims!(ax, 0, 1) + fig +end + +# ics3 = (q = ics.q[:, 1], p = ics.p[:, 1]) +# +# time_series2 = iterate(sympnet_nn, ics3; n_points = n_time_steps) +# +# function plot_solution3(time_step; theme = :light, framerate = 50) +# textcolor = theme == :dark ? :white : :black +# fig = Figure(size = (1000, 500), figure_padding = (5,50,5,5), fontsize = 24) +# ax = Axis(fig[1, 1], backgroundcolor = :transparent, +# bottomspinecolor = textcolor, +# topspinecolor = textcolor, +# leftspinecolor = textcolor, +# rightspinecolor = textcolor, +# xtickcolor = textcolor, +# ytickcolor = textcolor, +# xticklabelcolor = textcolor, +# yticklabelcolor = textcolor, +# xlabel=L"\omega", +# ylabel=L"q", +# xlabelcolor = textcolor, +# ylabelcolor = textcolor) +# lines!(ax, sol.s.q[time_step], label = rich("FOM + Implicit Midpoint"; color = textcolor), color = mblue) +# time_series = iterate(sympnet_nn, ics3; n_points = time_step) +# # prediction = (q = time_series.q[:, end], p = time_series.p[:, end]) +# prediction = (q = time_series2.q[:, time_step], p = time_series2.p[:, time_step]) +# prediction_big = decoder(sae_nn_cpu)(prediction) +# +# lines!(ax, prediction_big.q; label = rich("SAE + SympNet"; color = textcolor), color = mpurple) +# axislegend(ax; position = :rt) +# xlims!(ax, 0, 200) +# fig +# end + +# todo +#time_steps = 1:5 # axes(time_series.q, 2) +time_steps = 1:500 # axes(time_series.q, 2) + +for time_step in time_steps + fig1 = plot_solution(time_step) + save(sae_dir * "/sae-midpoint-$(string(time_step, pad = 3)).pdf", fig1) + + fig2 = plot_solution2(time_step) + save(sae_dir * "/sae-transformer-$(string(time_step, pad = 3)).pdf", fig2) + +# fig3 = plot_solution3(time_step) +# save(sae_dir * "/sae-sympnet-$(string(time_step, pad = 3)).pdf", fig3) +end diff --git a/scripts/symplectic_autoencoders/online_sympnet.jl b/scripts/symplectic_autoencoders/online_sympnet.jl index 096d4e79d..35a7d955f 100644 --- a/scripts/symplectic_autoencoders/online_sympnet.jl +++ b/scripts/symplectic_autoencoders/online_sympnet.jl @@ -26,8 +26,8 @@ sae_nn = NeuralNetwork(sae_arch, backend) const n_epochs = 262144 const batch_size = 4096 -sae_method = AdamOptimizerWithDecay(n_epochs) -o = Optimizer(sae_nn, sae_method) +sae_pairing = AdamOptimizerWithDecay(n_epochs) +o = Optimizer(sae_nn; sae_pairing...) println("Number of batches: ", GeometricMachineLearning.number_of_batches(dl, Batch(batch_size))) @@ -86,8 +86,8 @@ integrator_batch_size = 4096 seq_length = 4 integrator_architecture = StandardTransformerIntegrator(reduced_dim; transformer_dim = 10, n_blocks = 3, n_heads = 5, L = 2, upscaling_activation = tanh) integrator_nn = NeuralNetwork(integrator_architecture, backend) -integrator_method = AdamOptimizerWithDecay(integrator_train_epochs) -o_integrator = Optimizer(integrator_method, integrator_nn) +integrator_pairing = AdamOptimizerWithDecay(integrator_train_epochs) +o_integrator = Optimizer(integrator_nn; integrator_pairing...) loss = GeometricMachineLearning.ReducedLoss(encoder(sae_nn), decoder(sae_nn)) diff --git a/scripts/symplectic_autoencoders/online_transformer_for_sae.jl b/scripts/symplectic_autoencoders/online_transformer_for_sae.jl index 2f53e1973..eaa933bb1 100644 --- a/scripts/symplectic_autoencoders/online_transformer_for_sae.jl +++ b/scripts/symplectic_autoencoders/online_transformer_for_sae.jl @@ -24,8 +24,8 @@ const integrator_batch_size = 4096 const seq_length = 4 const integrator_architecture = StandardTransformerIntegrator(reduced_dim; transformer_dim = 20, n_blocks = 3, n_heads = 5, L = 3, upscaling_activation = tanh) const integrator_nn = NeuralNetwork(integrator_architecture, backend) -const integrator_method = AdamOptimizerWithDecay(integrator_train_epochs) -const o_integrator = Optimizer(integrator_method, integrator_nn) +const integrator_pairing = AdamOptimizerWithDecay(integrator_train_epochs) +const o_integrator = Optimizer(integrator_nn; integrator_pairing...) loss = GeometricMachineLearning.ReducedLoss(encoder(sae_nn), decoder(sae_nn)) diff --git a/scripts/volume_preserving_transformer/rigid_body.jl b/scripts/volume_preserving_transformer/rigid_body.jl index 589a368d5..e3ec12e8a 100644 --- a/scripts/volume_preserving_transformer/rigid_body.jl +++ b/scripts/volume_preserving_transformer/rigid_body.jl @@ -43,7 +43,7 @@ const dl = backend == CPU() ? DataLoader(dl₁.input) : DataLoader(dl₁.input | # hyperparameters concerning training const n_epochs = 500000 const batch_size = 16384 -const opt_method = AdamOptimizerWithDecay(n_epochs, T; η₁ = 1e-2, η₂ = 1e-6) +const opt_pairing = AdamOptimizerWithDecay(n_epochs, T; η₁ = 1e-2, η₂ = 1e-6) # parameters for evaluation ics_val = [sin(1.1), 0., cos(1.1)] @@ -54,7 +54,7 @@ const t_validation_long = 100 function train_the_network(nn₀::GeometricMachineLearning.NeuralNetwork, batch::Batch) Random.seed!(1234) - o₀ = Optimizer(opt_method, nn₀) + o₀ = Optimizer(nn₀; opt_pairing...) loss_array = o₀(nn₀, dl, batch, n_epochs) diff --git a/scripts/zygote.jl b/scripts/zygote.jl new file mode 100644 index 000000000..cde81e962 --- /dev/null +++ b/scripts/zygote.jl @@ -0,0 +1,20 @@ +using Zygote, Printf, LinearAlgebra + +const number_data_points = 1000 + +const data_input = [[i] for i in 1:number_data_points] + +function_to_be_differentiated(input, A) = norm(A*input) + +function gradient_eval(data, num, A = rand(100000,1)) + input = data[num] + @printf "First one: " + @time Zygote.gradient(A -> function_to_be_differentiated(input, A), A)[1] + @printf "Second one:" + @time Zygote.gradient(A -> function_to_be_differentiated(data[num], A), A)[1] + @printf "\n" +end + +for i in 1:5 + gradient_eval(data_input, Int(ceil(rand()*number_data_points))) +end From 0a6039fd82fb6c7095334b7615e18b68a18e810c Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Mon, 17 Aug 2026 21:59:00 +0900 Subject: [PATCH 05/12] Update the README example to the 0.5 optimizer interface The example predated the move of the optimizer machinery to GeometricOptimizers and no longer ran: - `Optimizer(AdamOptimizer(), g_nn)` becomes `Optimizer(Adam(type), g_nn; step_size = 1e-3)`. The method comes first and the learning rate is no longer part of it. `Adam` is constructed with the element type, so it reuses the `type` binding defined a few lines above -- which also makes the point that the example is type-generic. - `Iterate_Sympnet` is not defined in the package (`isdefined(GeometricMachineLearning, :Iterate_Sympnet) == false`), so the line raised an `UndefVarError` as written. `iterate` is what the SympNet tutorial uses. A paragraph says where the optimizer methods now come from, since the README is where a reader meets them first and `Adam` no longer being a GML type is otherwise unexplained. This is the content of #237, which was opened against the pre-CairoMakie README and whose plotting hunk #238 has since superseded. Closes #237. Co-Authored-By: Claude Opus 5 (1M context) --- README.md | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/README.md b/README.md index 7b4241b9c..dc693a09c 100644 --- a/README.md +++ b/README.md @@ -18,6 +18,8 @@ At its core every neural network comprises three components: a neural network ar Traditionally, physical properties have been encoded into the loss function (PINN approach), but in `GeometricMachineLearning.jl` this is exclusively done through the architectures and the optimizers of the neural network, thus giving theoretical guarantees that these properties are actually preserved. +The optimizer methods themselves — `GradientMethod`, `MomentumMethod`, `Adam`, the manifold types they act on, and the caches, global sections and retractions that go with them — come from [`GeometricOptimizers.jl`](https://github.com/JuliaGNI/GeometricOptimizers.jl) and are re-exported here. `GeometricMachineLearning.jl` supplies the part that is specific to neural networks: the architectures, the layers, and walking the parameter tree of a network during training. + Using the package is very straightforward and is very flexible with respect to the device `(CPU, CUDA, Metal, ...)` and the type `(Float16, Float32, Float64, ...)` you want to use. The following is a simple example to learn a SympNet on data coming from a pendulum: ```julia using GeometricMachineLearning @@ -41,8 +43,8 @@ backend = CUDABackend() # initialize the network (i.e. the parameters of the network) g_nn = NeuralNetwork(gsympnet, backend, type) -# call the optimizer -g_opt = Optimizer(AdamOptimizer(), g_nn) +# call the optimizer: the method comes first, the step size is given separately +g_opt = Optimizer(Adam(type), g_nn; step_size = 1e-3) const nepochs = 300 const batch_size = 100 @@ -53,7 +55,7 @@ g_loss_array = g_opt(g_nn, dl, Batch(batch_size), nepochs) # plot the result ics = (q=qp_data.q[:,1], p=qp_data.p[:,1]) const steps_to_plot = 200 -g_trajectory = Iterate_Sympnet(g_nn, ics; n_points = steps_to_plot) +g_trajectory = iterate(g_nn, ics; n_points = steps_to_plot) fig = Figure() ax = Axis(fig[1, 1]; xlabel = "q", ylabel = "p") lines!(ax, vec(qp_data.q')[1:steps_to_plot], vec(qp_data.p')[1:steps_to_plot]; label = "training data") From 77e7d06297807dcf321f0340c9f156d6c29bbd1c Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Mon, 17 Aug 2026 21:59:04 +0900 Subject: [PATCH 06/12] Require GeometricOptimizers 0.4, and record the release The `[sources]` entries that pointed `GeometricOptimizers` at a sibling checkout are gone from `Project.toml` and `docs/Project.toml`, so both environments resolve it from the General registry -- `registries = "General"`, no `path` key. 0.4.0 is where the interface GML now imports became public API, so `[compat]` is `"0.4"` and GML does not load against 0.3. Removing them is not `Pkg.free`, which fails here with "could not find source path for package GeometricOptimizers", and not `Pkg.resolve` either -- that infers `[sources]` from the manifest and writes the entry straight back. Delete the two entries, delete the (gitignored) manifests, `Pkg.instantiate`. `docs/inventories/GeometricOptimizers.toml` is regenerated from the *deployed* 0.4.0 documentation rather than from a local build. It had drifted by one heading -- a trailing period removed upstream after the file was first generated -- which nothing here linked to, but a stale inventory is a dead link waiting for someone to add the reference. The changelog gains two entries under *Open Issues* that were being tracked outside it: C10 ten exported names are undefined, measured with `[n for n in names(GML) if !isdefined(GML, n)]`. This release removed three of the thirteen; the rest each need a decision, and upstream's `test/exports.jl` shows what closes the class. C11 41 test files are unreachable from `runtests.jl`. Not one problem but three -- GPU tests, performance probes, and the `train!` suite that B6 says is broken -- which is why the entry asks for a decision per group rather than a deletion. The `[Unreleased]` heading stays until the release is tagged. The version is `0.5.0`. Co-Authored-By: Claude Opus 5 (1M context) --- CHANGELOG.md | 265 +++++++--- Project.toml | 4 +- docs/Project.toml | 1 - docs/inventories/GeometricOptimizers.toml | 570 +++++++++++----------- 4 files changed, 478 insertions(+), 362 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index 453def7cc..fbf7ec083 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -17,17 +17,86 @@ breaking release). **The optimizer machinery moves to [GeometricOptimizers][go].** GML no longer implements its own optimizers: the methods, caches, states, global sections and retractions all come from -GeometricOptimizers v0.2, and GML keeps only the parts that are about neural networks — walking a +GeometricOptimizers, and GML keeps only the parts that are about neural networks — walking a `NeuralNetworkParameters` tree, and the manifold layer types. +**Requires GeometricOptimizers 0.4.** The eleven geometry types GML used to define itself are +`import`ed from there now, and the interface it needs — `metric`, `check`, `Ω`, `global_section`, +`apply_section!`, `update_section!`, the retraction types, `AdamOptimizerWithDecay` — only became +public API in [GeometricOptimizers 0.4.0][go50]. GML does not load against 0.3. + This is a breaking release and the break is not mechanical. Read *Changed (breaking)* before upgrading. [go]: https://github.com/JuliaGNI/GeometricOptimizers.jl [go45]: https://github.com/JuliaGNI/GeometricOptimizers.jl/pull/45 +[go50]: https://github.com/JuliaGNI/GeometricOptimizers.jl/pull/50 ### Removed (breaking) +- **GML's copies of eleven types GeometricOptimizers also defines.** `Manifold`, `StiefelManifold`, + `GrassmannManifold`, `SkewSymMatrix`, `SymmetricMatrix`, `AbstractTriangular`, `LowerTriangular`, + `UpperTriangular`, `AbstractLieAlgHorMatrix`, `StiefelLieAlgHorMatrix`, + `GrassmannLieAlgHorMatrix` and `StiefelProjection` are now *imported* from GeometricOptimizers and + re-exported. Twelve files go with them — all of `src/arrays/` bar `poisson_tensor.jl`, all of + `src/manifolds/`, and `src/optimizers/go_bridges.jl` — about 2500 lines. + + The copies were near-verbatim, but Julia saw them as *distinct types*, so none of + GeometricOptimizers' generic machinery dispatched on them: GML re-implemented + `geodesic`, `cayley`, `apply_section!`, `global_rep` and `update_section!` once per manifold, and + `go_bridges.jl` held some thirty more methods reconnecting the two hierarchies. All of that is + gone. This closes **B2** + ([#234](https://github.com/JuliaGNI/GeometricMachineLearning.jl/issues/234)). + + `import` rather than `const X = GeometricOptimizers.X`: GML adds constructor methods to several of + these types, and extending a type reached through `using` warns on every such method since Julia + 1.12. + + Not a source break for a caller — the names are still exported and mean the same thing — but the + *types* are now GeometricOptimizers', so `x isa GeometricMachineLearning.StiefelManifold` and + `x isa GeometricOptimizers.StiefelManifold` are the same question, where before they were + different ones with different answers. + +- **`AdamOptimizerWithDecay` is GeometricOptimizers'**, and GML's own is deleted. This closes **B1**: + both packages exported the name, so `using GeometricMachineLearning, GeometricOptimizers` failed + outright on it. It was the same algorithm — Adam's direction with a learning rate decaying by the + same `γ = exp(log(η₂/η₁)/n)` — packaged differently, and upstream's packaging is the right one: + the direction is an `Adam` method and the schedule is a `DecayingStatic` line search. + + **What a call has to change.** It is now a `(algorithm, linesearch)` pairing rather than an + `OptimizerMethod`, so it splats into `Optimizer` instead of being passed positionally, `T` is + positional and defaults to `Float64` rather than being taken from `η₁` (so `Float32`), and the + moment coefficients are the keywords `β₁`, `β₂` rather than positional `ρ₁`, `ρ₂`: + + ```julia + Optimizer(AdamOptimizerWithDecay(n_epochs), nn) # before + Optimizer(nn; AdamOptimizerWithDecay(n_epochs, Float32)...) # after + ``` + +- **The optimizer caches stop being exported.** `AbstractCache`, `GradientCache`, `MomentumCache` + and `AdamCache`. They are `solver_step!` scratch and stay internal to GeometricOptimizers, for + every method alike; reach one as `GeometricOptimizers.AdamCache` if you genuinely need to name it. + +- **`update!` stops being exported.** GML imported `AbstractNeuralNetworks.update!` and never added a + method to it, so all the export did was shadow `GeometricOptimizers.update!` — a *different* + generic function, and the one that actually has methods for the optimizer caches. That one is + re-exported now instead. + +- **`SymplecticLieAlgMatrix`, `SymplecticLieAlgHorMatrix` and `SymplecticProjection` stop being + exported.** Nothing has defined them for as long as the git history goes back, so the exports were + silent `UndefVarError`s waiting for a caller. + +- **Twelve test files that duplicated GeometricOptimizers' suite**, under `test/arrays/`, + `test/manifolds/` and `test/optimizers/utils/`. They tested the shared types, which upstream tests + itself; what they covered and upstream did not was ported there first (see its changelog — it + turned up four defects in the upstream suite, including a test file that never tested the Stiefel + global section). `test/arrays/triangular.jl` keeps the half that tests GML's tensor kernels. + + Eight further files went with them — `test/optimizers/{exponential_retractions, riemannian_gradients, + hor_lift, lie_alg_lifts, manifold_optim, momentum_optim_test, standard_optim_test}.jl` and + `test/optimizers/manifold_related/legacy_functions.jl`. All were unreachable from `runtests.jl`, + and most could not have run: two `include` paths deleted years ago, three `using Lux`. + - **`BFGSOptimizer` and `BFGSCache`**, along with `docs/src/optimizers/bfgs_optimizer.md`. This entry used to say that GeometricOptimizers "has `_BFGS()` and its own cache" and that GML's @@ -41,11 +110,16 @@ upgrading. | fits GML's per-leaf tree update? | yes — that is what it was for | no | `_is_go_native_method` therefore sends `BFGS` down GML's Euclidean path, where - `_euclidean_update!` has no method for it and the step raises a `MethodError`. Bridging it needs - `_fill!`, `_difference!`, `outer!` and the `ParameterHandling.flatten` round-trip taught about - GML's manifold and lift types — GML's `StiefelManifold` is a *different type* from - GeometricOptimizers', and the two hierarchies are unrelated, so none of GO's `Manifold` methods - apply. That work is not done. + `_euclidean_update!` has no method for it and the step raises a `MethodError`. That work is not + done. + + It did get cheaper, though. This entry used to add that bridging it needs `_fill!`, + `_difference!`, `outer!` and the `ParameterHandling.flatten` round-trip *taught about GML's + manifold and lift types*, because those were different types from GeometricOptimizers' and none of + its `Manifold` methods applied. After the type unification above that half is simply gone — + upstream's `flatten`, `_fill!`, `_difference!` and `outer!` already work on these types, because + they are now the same types. What remains is routing `BFGS` through the per-leaf tree update at + all, which is the same question as **C1**. Until it is, use `AdamOptimizer()`, `MomentumOptimizer()` or `GradientOptimizer()`. - **`SymplecticStiefelManifold`.** Never reachable — the file that defined it was commented out of @@ -167,8 +241,63 @@ continuation lines, and reading it misses them. `scripts/` as though they were current, and `hnn/` predates two generations of the optimizer and architecture APIs. +### Documentation + +- **The `Manifolds` and `Optimizer` chapters move to GeometricOptimizers**, together with the two + `Special Arrays and AD` pages whose data structures are its — `arrays/skew_symmetric_matrix.md` and + `arrays/global_tangent_spaces.md`. Thirteen pages, ~3050 lines, documenting types that live there + now. `arrays/tensors.md` and `pullbacks/computation_of_pullbacks.md` stay: they document GML's own + tensor kernels and AD. + + `optimizers/optimizer_framework.md` splits. Its framework theory merges into upstream's + `manifold_optimizers.md`; what is left is a new `optimizers/optimizer.md` covering GML's own + `Optimizer`, `optimize_for_one_epoch!` and `optimization_step!` — the parameter tree and the + training loop. + +- **What the PDF book loses.** `_latex_pages` drops the whole `Background → Manifolds` chapter and + the four-page `Optimizers` part, keeping a one-page `Optimizer` chapter, and the Appendix's + `Special Arrays, Tensors and Pullbacks` becomes `Tensors and Pullbacks`. The book now opens on + geometric structure and takes the manifold optimizers as given, citing them. + +- **`DocumenterInterLinks`** enters `docs/Project.toml` and `docs/make.jl`, with a committed + inventory under `docs/inventories/`. Thirty-six references from the chapters that stayed into the + ones that moved are now real cross-references rather than dangling `@ref`s, and the seven + de-referenced code spans C3 complained about (`𝔄`, `cayley`, `update!` …) can be links again. + This closes **C3**. + ### Fixed +- **`Matrix + SkewSymMatrix` was a `StackOverflowError`.** `Base.:+(B::AbstractMatrix, + A::SkewSymMatrix)` read `B + A`, which is itself. Fixed by the type unification above: upstream's + method, which reads `A + B`, has always been right. GeometricOptimizers' suite now asserts that + addition against a dense matrix commutes, for all four structured types rather than for the one + instance. + +- **`parent(::StiefelLieAlgHorMatrix)` referenced an unbound variable.** It returned `(A, B)` where + `B` was never defined — an `UndefVarError` for any caller. Also fixed by the unification; + upstream returns `(A.A, A.B)`, which is what its `vec(::AbstractLieAlgHorMatrix)` builds on. + +- **A decaying step size was read one step early.** `optimization_step!` read the step size *before* + incrementing `opt.iterations`, so the first step of a run took `α(0) = η₁` where the pre-0.5 + `AdamOptimizerWithDecay` incremented first and took `α(1) = γη₁`. Every step of a run was therefore + one place early in the schedule. The increment now comes first, which is also how + `DecayingStatic` counts and how `GeometricOptimizers.solve!` counts (it calls + `increase_iteration_number!` before `solver_step!`) — and what upstream's + `test/adam_optimizer_with_decay.jl` asserts GML does. Pinned by `schedule_starts_at_one` in + `test/optimizers/optimizer_convergence_tests/adam_with_learning_rate_decay.jl`. + + It affected only a decaying step size; a fixed one is the same at every `t`. + +- **A pullback test asserted nothing.** The loop in `test/arrays/triangular.jl` comparing the batched + `mat_tensor_mul` pullback against the single-slice one was written as bare expressions rather than + `@test`s, so it ran and discarded its results. They are `@test`s now, and they pass. + +- **`solve!` was a second generic function.** GML's `solve!(::NeuralNetwork{<:PSDArch}, …)` — solve + for the parameters directly, by SVD, rather than training for them — created a new function of that + name rather than adding a method to the one a caller already had. It is imported from + GeometricOptimizers now, so `using GeometricMachineLearning, GeometricOptimizers` no longer + collides on it either. + - **The optimizer path no longer takes ten hours to compile through a function.** Inference spun in method-table intersection whenever `GeometricOptimizers.update!` was reached through GML's optimizer tree, and it produced no error — CI showed jobs running past 1 h 15 min against ~25–48 @@ -456,62 +585,6 @@ they resolved to is in the release notes above. ### B. Known defects -- **B1. Both packages export `AdamOptimizerWithDecay`.** GeometricOptimizers v0.2.0 ships the name - and GML defines and exports its own (`src/optimizers/optimizer.jl:35`). GML itself loads, but - `using GeometricMachineLearning, GeometricOptimizers` in downstream code fails outright: - - ``` - UndefVarError: `AdamOptimizerWithDecay` not defined in `Main` - Hint: It looks like two or more modules export different bindings with this name… - ``` - - Deleting GML's copy is the fix, but it is **not** independent of C1: GeometricOptimizers' - `AdamOptimizerWithDecay(n, T; …)` returns an `(algorithm, linesearch)` pairing for its own - `Optimizer(x, problem; method...)`, whereas GML's `Optimizer` carries a scalar `step_size` and - computes the schedule in `_current_step_size`. Un-exporting it would clear the ambiguity on its - own if a stopgap is wanted first. - -- **B2. `Manifold` is split between the two packages - ([#234](https://github.com/JuliaGNI/GeometricMachineLearning.jl/issues/234)).** GML's - `StiefelManifold` and `GrassmannManifold` subtype `GeometricMachineLearning.Manifold`, which is - distinct from `GeometricOptimizers.Manifold`, so GeometricOptimizers' generic `geodesic` and - `cayley` never dispatch on them. Commit `b16267ea` worked around this by re-implementing the - pipeline four times, with the same duplication on the Lie-algebra-horizontal types and the - `_copyto!`/`_add!`/`_rac!`/`_square!`/`_div!` family — about thirty bridge methods. - - The decided fix is `const Manifold = GeometricOptimizers.Manifold`, which lets all four bridge - methods be deleted. It is not a one-liner: `src/manifolds/abstract_manifold.jl` is a near-verbatim - copy of GeometricOptimizers', so after aliasing, GML's generic methods would have *identical* - signatures to the upstream ones and silently overwrite them — a hard precompilation error on Julia - ≥ 1.13. GML's copies have to go in the same change, keeping only what genuinely differs. - - The `_gml_rgrad` bug fixed in this release was the same split showing up somewhere it changed - results silently, which is an argument for closing this sooner rather than later. - -- **B4. The documentation's executable examples still use the old optimizer internals.** The - Documentation and PDF builds get past resolution and doctests now and fail in the `@example` - blocks — 32 errors across 12 pages, because the documentation teaches the optimizer by reaching - into its cache, and GeometricOptimizers' caches are not shaped like GML's were: - - ``` - type MomentumCache has no field `A`, available fields: `x`, `g`, `δ`, `Δg`, `g̃`, `g̃_is_current`, `section` - type AdamCache has no field `Y` - no method matching update!(::Optimizer{GradientMethod, GradientCache{…}}, …) - `update_section!` not defined - no method matching AdamCache(::@NamedTuple{weight::SymmetricMatrix{Float16, …}}, …) - ``` - - `optimizer_methods.md` is the bulk of it: eleven call sites building `dx = (A = one(weight.A),)`, - calling `update!(o, o.cache, dx)` and printing `o.cache.A` or `o.cache.Y` to show what a cache - holds. `parallel_transport.md` uses `update_section!`, which GML no longer exports. The three - named `@example` blocks that fail outright are `sympnet`, `rigid_body` and `s2_parallel_transport`. - - This is not a mechanical rename. The pages explain how the optimizer works by showing its - internals, and those internals now belong to another package with a different design — so closing - this means deciding how much of that exposition GML's documentation should still carry, and - rewriting it against the upstream API or handing it to GeometricOptimizers' own documentation. It - is the last thing standing between this branch and green Documentation and PDF workflows. - - **B5. The symbolic pullback of `HNNLoss` is not the gradient of the batched loss.** `SymbolicPullback` differentiates the loss of a *single* sample and sums the per-sample gradients (`reduce = +`), which equals the gradient of the batched loss only when the loss is a sum over @@ -552,26 +625,31 @@ they resolved to is in the release notes above. a `NeuralNetwork{<:HamiltonianArchitecture}` — `hamiltonian_vector_field` is the obvious candidate — and giving it a test that runs. - (There is no **B3** any more — it was `input_dimension`/`output_dimension` existing twice, closed - by SymbolicNeuralNetworks 0.5; see *Fixed* above. The number is left vacant rather than reused.) + (**B1**, **B2**, **B3** and **B4** are all closed and their entries are gone: B1 and B2 by this + release — the duplicated `AdamOptimizerWithDecay` and the split `Manifold`, both under *Removed + (breaking)* — B3 by SymbolicNeuralNetworks 0.5, and B4 by `a427add1`, which repaired the + documentation build. The numbers are left vacant rather than reused.) ### C. Follow-ups and cleanups -- **C1. The rest of the optimizer machinery still belongs upstream.** The traversal - (`_make_optimizer_cache`, `_make_optimizer_state`, `_tree_optim_step!`, `_leaf_optim_step!`) and - the bespoke `GMLEuclideanState` are GML implementations of what GeometricOptimizers v0.2 already - supports natively. Route: branch GeometricOptimizers, move it, delete it here, open an issue - referencing the upstream PR. B1 unblocks with it. +- **C1. The parameter-tree traversal still belongs upstream.** `_make_optimizer_cache`, + `_make_optimizer_state`, `_tree_optim_step!`, `_leaf_optim_step!` and the bespoke + `GMLEuclideanState` are GML implementations of what GeometricOptimizers supports natively for a + single parameter. `GMLEuclideanState` in particular duplicates what `GradientState`, + `MomentumState` and `AdamState` already do for a plain array. + + What has to go upstream is *not* a reuse of GeometricOptimizers' `Optimizer`: that one needs an + `OptimizerProblem`, i.e. an objective function, and minibatch training has none — the gradient + arrives from AD one batch at a time. It is a new entry point there, a + gradient-supplied-externally step over a `NamedTuple` parameter tree. GML's `Optimizer` would then + be the `NeuralNetwork` constructor and the training-loop functor, and nothing else. -- **C2. Two `isa` branches remain in `_leaf_optim_step!`** (`src/optimizers/optimizer.jl:182`, - `:187`, for `AdamState`/`MomentumState`). Measurement showed the traversal is not implicated in - the compile-time problem, so this is tidying, and it disappears entirely if C1 lands first. + `Optimizer` is the one name still exported by both packages, so this is also what closes the last + of B1's class of collision. -- **C3. Cross-package documentation links are prose, not links.** The seven `@ref`s fixed above were - de-referenced into plain code spans, which is the cheap fix rather than the right one. - `DocumenterInterLinks` would let GML's documentation link into GeometricOptimizers' properly, so - that `𝔄`, `cayley` and `update!` become real cross-references again. GeometricOptimizers' own - `docs/Project.toml` already carries it; GML's does not. +- **C2. Two `isa` branches remain in `_leaf_optim_step!`** (for `AdamState`/`MomentumState`). + Measurement showed the traversal is not implicated in the compile-time problem, so this is tidying, + and it disappears entirely if C1 lands first. - **C4. `[compat]` entries worth revisiting.** `ForwardDiff = "0.10, 1"` is dead weight — GeometricOptimizers requires 1, so the resolver picks it regardless and the `0.10` branch is @@ -611,6 +689,39 @@ they resolved to is in the release notes above. (it generated the pendulum training data, which `scripts/pendulum.jl` now does) or delete the scripts that need it. +- **C10. Ten exported names are undefined.** `CPUDevice`, `Device`, `LinearSymplecticLayerP`, + `LinearSymplecticLayerQ`, `ResidualLayer`, `aresame`, `convert_to_dev`, `description`, `symbol` + and `timestep` are in an `export` list and defined nowhere, so + `[n for n in names(GeometricMachineLearning) if !isdefined(GeometricMachineLearning, n)]` returns + all ten. They are harmless in the sense that nothing breaks until someone reaches for one, at which + point they get `UndefVarError` from a name the package advertises. + + This release removed the three that happened to sit in the export block it was already rewriting + (`SymplecticLieAlgMatrix`, `SymplecticLieAlgHorMatrix`, `SymplecticProjection`), which is why the + count is ten rather than thirteen. The rest are spread across the module and were left alone + deliberately: each needs a decision — define it, or drop the export — and a few are load-bearing + names in prose (`description` is `export`ed with the comment "from GeometricBase to print docs", + and GeometricBase does define it, so that one is likely an `import` that was never written). + + `GeometricOptimizers`' `test/exports.jl` closes this whole class with one assertion over `names`; + this package has no equivalent, and adding one is the actual fix. + +- **C11. 41 test files are unreachable from `runtests.jl`.** By area: 20 under `performance_tests/`, + 5 `orthogonalization_procedures/`, 4 `train!/`, 2 `cuda/`, and 10 singletons (`training_phnn.jl`, + `macro_testerror.jl`, `integrator/test_integrator.jl`, `attention_layer/`, `custom_ad_rules/`, + `data/`, `kernels/`, `layers/`, `symplectic_autoencoders/`, `transformer_related/`). + + They are not all the same thing, which is why this is one issue and not a deletion. The + `performance_tests/` and `cuda/` files need hardware the suite does not assume; the `train!/` files + cover `train!`, which **B6** says is broken for every method, so they would fail if enabled; and + the singletons are mostly stale. What they have in common is that nothing runs them, so nothing + tells you when they rot — `test/optimizers/lie_alg_lifts.jl`, deleted in this release, had been + including `../src/arrays/skew_sym.jl` since before that path stopped existing. + + This release deleted the eight that were `GeometricOptimizers` material *and* could not have run. + The remainder needs a decision per group: register them behind an environment flag (the GPU and + performance ones), fix the thing they test (`train!`), or delete them. + ### D. Unverified Not defects — claims this release makes that nothing has actually checked yet. diff --git a/Project.toml b/Project.toml index 9c6e3091c..fbc33712e 100644 --- a/Project.toml +++ b/Project.toml @@ -1,6 +1,6 @@ name = "GeometricMachineLearning" uuid = "194d25b2-d3f5-49f0-af24-c124f4aa80cc" -version = "0.5.0-DEV" +version = "0.5.0" authors = ["Michael Kraus "] [deps] @@ -41,7 +41,7 @@ ForwardDiff = "0.10, 1" GeometricBase = "0.14" GeometricEquations = "0.21" GeometricIntegrators = "0.18.2" -GeometricOptimizers = "0.2.1" +GeometricOptimizers = "0.4" GeometricSolutions = "0.6" HDF5 = "0.16, 0.17" KernelAbstractions = "0.9" diff --git a/docs/Project.toml b/docs/Project.toml index b12c8ba9f..d1ca49661 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -19,4 +19,3 @@ Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" [sources] BrenierTwoFluid = {rev = "main", url = "https://github.com/ToBlick/BrenierTwoFluids.git"} GeometricMachineLearning = {path = ".."} -GeometricOptimizers = 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unrelated decays" name = "Two-unrelated-decays" -uri = "weight_decay.html#$" +uri = "weight_decay/#$" [[std.label]] dispname = "Using them" name = "Using-them" -uri = "retractions.html#$" +uri = "retractions/#$" [[std.label]] dispname = "Vector Fields" name = "Vector-Fields" -uri = "manifolds/manifolds.html#$" +uri = "manifolds/manifolds/#$" [[std.label]] dispname = "Weight Decay on Manifolds" name = "Weight-Decay-on-Manifolds" -uri = "weight_decay.html#$" +uri = "weight_decay/#$" [[std.label]] dispname = "Weights on Manifolds" name = "Weights-on-Manifolds" -uri = "optimizer_methods.html#$" +uri = "optimizer_methods/#$" [[std.label]] dispname = "What a retraction is" name = "What-a-retraction-is" -uri = "retractions.html#$" +uri = "retractions/#$" [[std.label]] dispname = "What the argument does and does not depend on" name = "What-the-argument-does-and-does-not-depend-on" -uri = "weight_decay.html#$" +uri = "weight_decay/#$" [[std.label]] dispname = "What they cost" name = "What-they-cost" -uri = "retractions.html#$" +uri = "retractions/#$" [[std.label]] dispname = "What they cost and how accurate they are" name = "What-they-cost-and-how-accurate-they-are" -uri = "retractions.html#$" +uri = "retractions/#$" [[std.label]] dispname = "Where GeometricMachineLearning's method belongs" name = "Where-GeometricMachineLearning's-method-belongs" -uri = "weight_decay.html#Where-GeometricMachineLearning%27s-method-belongs" +uri = "weight_decay/#Where-GeometricMachineLearning%27s-method-belongs" [[std.label]] dispname = "Where a retraction sits in the algorithm" name = "Where-a-retraction-sits-in-the-algorithm" -uri = "retractions.html#$" +uri = "retractions/#$" [[std.label]] dispname = "Which line search" name = "Which-line-search" -uri = "weight_decay.html#$" +uri = "weight_decay/#$" [[std.label]] dispname = "Why the decay vanishes" name = "Why-the-decay-vanishes" -uri = "weight_decay.html#$" +uri = "weight_decay/#$" [[std.label]] dispname = "Why the name is not AdamW" name = "Why-the-name-is-not-AdamW" -uri = "weight_decay.html#$" +uri = "weight_decay/#$" [[std.label]] dispname = "Why the storage matters here" name = "Why-the-storage-matters-here" -uri = "special_matrices.html#$" +uri = "special_matrices/#$" From dfa5b55a51961df49dc101f4f14e62fd42c982c8 Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Mon, 17 Aug 2026 23:19:35 +0900 Subject: [PATCH 07/12] Take the scratch scripts back out of the repository `ed30aee8` was about bringing the existing scripts to the new `AdamOptimizerWithDecay`, but it also added four files that were untracked in the working tree at the time: `enzyme.jl`, `zygote.jl`, `sae_script2.jl` and `harmonic_oscillator_sympnet.jl`. Nothing in the commit message, the PR description or the changelog mentions them, and they do not look like they were meant to be committed -- `enzyme.jl` needs `Enzyme`, which is not in `scripts/Project.toml`, and `sae_script2.jl` reads `../docs/src/tutorials/*.jld2` by relative path and writes a thousand PDFs. Co-Authored-By: Claude Opus 5 (1M context) --- scripts/enzyme.jl | 39 ------ scripts/harmonic_oscillator_sympnet.jl | 58 -------- scripts/sae_script2.jl | 181 ------------------------- scripts/zygote.jl | 20 --- 4 files changed, 298 deletions(-) delete mode 100644 scripts/enzyme.jl delete mode 100644 scripts/harmonic_oscillator_sympnet.jl delete mode 100644 scripts/sae_script2.jl delete mode 100644 scripts/zygote.jl diff --git a/scripts/enzyme.jl b/scripts/enzyme.jl deleted file mode 100644 index 78612cd4a..000000000 --- a/scripts/enzyme.jl +++ /dev/null @@ -1,39 +0,0 @@ -using BenchmarkTools -using Enzyme -using LinearAlgebra -using Zygote - - -loss(A,x) = norm(A*x) - -# function loss(A,x) -# y = zero(x) -# mul!(y,A,x) -# norm(y) -# end - -function test(n) - A = rand(n,n) - x = rand(n) - - l = a -> loss(a,x) - - dA = zero(A) - - println("\nn = $n") - - println("\nEnzyme (autodiff):") - @btime Enzyme.autodiff(Reverse, $l, Active, Duplicated($A, $dA)) - - println("\nEnzyme (gradient):") - @btime Enzyme.gradient(Reverse, $l, $A) - - println("\nZygote:") - @btime Zygote.gradient($l, $A)[1] - - println("") -end - -test(100) -test(1000) -test(10000) diff --git a/scripts/harmonic_oscillator_sympnet.jl b/scripts/harmonic_oscillator_sympnet.jl deleted file mode 100644 index 4f48ec5d6..000000000 --- a/scripts/harmonic_oscillator_sympnet.jl +++ /dev/null @@ -1,58 +0,0 @@ -using GeometricMachineLearning -using GeometricIntegrators: ImplicitMidpoint, integrate -import GeometricProblems.HarmonicOscillator as ho - -# the problem is the ODE of the harmonic oscillator -ho_problem = ho.hodeproblem(; tspan = 500) - -# integrate the system -solution = integrate(ho_problem, ImplicitMidpoint()) - -dl_raw = DataLoader(solution; suppress_info = true) - -# specify the data type and the backend -type = Float64 -backend = CPU() - -# we can then make a new instance of `DataLoader` with this backend and type. -dl = DataLoader(dl_raw, backend, type) - - -const upscaling_dimension = 2 -const nhidden = 1 -const activation = tanh -const n_layers = 4 # number of layers for the G-SympNet -const depth = 4 # number of layers in each linear block in the LA-SympNet - -# calling G-SympNet architecture -gsympnet = GSympNet(dl; upscaling_dimension = upscaling_dimension, - n_layers = n_layers, - activation = activation) - -# initialize the networks -g_nn = NeuralNetwork(gsympnet, backend, type) - -# set up optimizer; for this we first need to specify the optimization method -opt_method = AdamOptimizer(type) - -# we then call the optimizer struct which allocates the cache -g_opt = Optimizer(opt_method, g_nn) - -# determine the batch size (the number of samples in one batch) -const batch_size = 16 - -batch = Batch(batch_size) - -# number of training epochs -const nepochs = 100 - -# perform training (returns array that contains the total loss for each training step) -g_loss_array = g_opt(g_nn, dl, batch, nepochs; show_progress = false) - -ics = (q=dl.input.q[:, 1, 1], p=dl.input.p[:, 1, 1]) - -steps_to_plot = 1000 - -#predictions -g_trajectory = iterate(g_nn, ics; n_points = steps_to_plot) - diff --git a/scripts/sae_script2.jl b/scripts/sae_script2.jl deleted file mode 100644 index 902a8db61..000000000 --- a/scripts/sae_script2.jl +++ /dev/null @@ -1,181 +0,0 @@ -using GeometricIntegrators: integrate, ImplicitMidpoint -using GeometricMachineLearning -import Random # hide -import GeometricProblems.TodaLattice as tl -using JLD2 -using CairoMakie - -sae_dir = "animations" -mkpath(sae_dir) - -N = tl.Ñ # hide -Δx = 1. / (N - 1) # hide -Ω = -0.5 : Δx : 0.5 # hide -tl.μ - -# todo -#pr = tl.hodeproblem(; tspan = (0.0, 8.)) -pr = tl.hodeproblem(; tspan = (0.0, 800.)) -@time "FOM + Implicit Midpoint" sol = integrate(pr, ImplicitMidpoint()) - -dl_cpu = DataLoader(sol; autoencoder = true, suppress_info = true) - -const reduced_dim = 2 - -Random.seed!(123) # hide -sae_arch = SymplecticAutoencoder(dl_cpu.input_dim, reduced_dim; n_encoder_blocks = 4, - n_decoder_blocks = 4, - n_encoder_layers = 2, - n_decoder_layers = 2) - -const mtc = GeometricMachineLearning.map_to_cpu - -sae_trained_parameters = load("../docs/src/tutorials/sae_parameters.jld2")["sae_parameters"] -_nnp(ps::Tuple) = NeuralNetworkParameters{Tuple(Symbol("L$(i)") for i in 1:length(ps))}(ps) -sae_nn_cpu = NeuralNetwork(sae_arch, Chain(sae_arch), _nnp(sae_trained_parameters), CPU()) - -sae_rs = HRedSys(pr, encoder(sae_nn_cpu), decoder(sae_nn_cpu); integrator = ImplicitMidpoint()) - -# @time "FOM + Implicit Midpoint" sol_full = integrate_full_system(sae_rs) # hide -@time "SAE + Implicit Midpoint" sol_sae_reduced = integrate_reduced_system(sae_rs) # hide - - - - - -const T = Float32 -_T(qp::NamedTuple{(:q, :p)}) = (q = T.(qp.q), p = T.(qp.p)) - -dl_reduced = DataLoader(encoder(sae_nn_cpu)(_T(dl_cpu.input))) - -# lines(dl_reduced.input.q[1, :, 1], dl_reduced.input.p[1, :, 1]) - -# sympnet_arch = GSympNet(2; n_layers = 10) -# sympnet_nn = NeuralNetwork(sympnet_arch, T) -# o = Optimizer(AdamOptimizer(), sympnet_nn) -# o(sympnet_nn, dl_reduced, Batch(10), 500) - -morange = RGBf(255 / 256, 127 / 256, 14 / 256) -mred = RGBf(214 / 256, 39 / 256, 40 / 256) -mpurple = RGBf(148 / 256, 103 / 256, 189 / 256) -mblue = RGBf(31 / 256, 119 / 256, 180 / 256) -mgreen = RGBf(44 / 256, 160 / 256, 44 / 256) - -function plot_solution(time_step; theme = :light, framerate = 50) - textcolor = theme == :dark ? :white : :black - fig = Figure(size = (1000, 500), figure_padding = (5,50,5,10), fontsize = 24) - ax = Axis(fig[1, 1], backgroundcolor = :transparent, - bottomspinecolor = textcolor, - topspinecolor = textcolor, - leftspinecolor = textcolor, - rightspinecolor = textcolor, - xtickcolor = textcolor, - ytickcolor = textcolor, - xticklabelcolor = textcolor, - yticklabelcolor = textcolor, - xlabel=L"\omega", - ylabel=L"q", - xlabelcolor = textcolor, - ylabelcolor = textcolor) - lines!(ax, sol.s.q[time_step], label = rich("FOM + Implicit Midpoint"; color = textcolor), color = mblue) - lines!(ax, sae_rs.decoder((q = sol_sae_reduced.s.q[time_step], p = sol_sae_reduced.s.p[time_step])).q, label = rich("SAE + Implicit Midpoint"; color = textcolor), color = mgreen) - axislegend(ax; position = :rt) - xlims!(ax, 0, 200) - ylims!(ax, 0, 1) - fig -end - - -#### Transformer - -const seq_length = 4 -integrator_architecture = StandardTransformerIntegrator(reduced_dim; - transformer_dim = 20, - n_blocks = 3, - n_heads = 5, - L = 3, - upscaling_activation = tanh) - -nn_integrator_parameters = load("../docs/src/tutorials/integrator_parameters.jld2")["integrator_parameters"] # hide -integrator_nn = NeuralNetwork(integrator_architecture, Chain(integrator_architecture), _nnp(nn_integrator_parameters), CPU()) # hide - -# todo -#n_time_steps = 100 -n_time_steps = 10000 - -ics = (q = dl_reduced.input.q[:, 1:seq_length], p = dl_reduced.input.p[:, 1:seq_length]) -time_series = iterate(mtc(integrator_nn), ics; n_points = n_time_steps, prediction_window = seq_length) -function plot_solution2(time_step; theme = :light, framerate = 50) - textcolor = theme == :dark ? :white : :black - fig = Figure(size = (1000, 500), figure_padding = (5,50,5,10), fontsize = 24) - ax = Axis(fig[1, 1], backgroundcolor = :transparent, - bottomspinecolor = textcolor, - topspinecolor = textcolor, - leftspinecolor = textcolor, - rightspinecolor = textcolor, - xtickcolor = textcolor, - ytickcolor = textcolor, - xticklabelcolor = textcolor, - yticklabelcolor = textcolor, - xlabel=L"\omega", - ylabel=L"q", - xlabelcolor = textcolor, - ylabelcolor = textcolor) - lines!(ax, sol.s.q[time_step], label = rich("FOM + Implicit Midpoint"; color = textcolor), color = mblue) - # prediction = (q = time_series.q[:, end], p = time_series.p[:, end]) - prediction = (q = time_series.q[:, time_step], p = time_series.p[:, time_step]) - prediction_big = decoder(sae_nn_cpu)(prediction) - - lines!(ax, prediction_big.q; label = rich("SAE + Transformer"; color = textcolor), color = mpurple) - axislegend(ax; position = :rt) - xlims!(ax, 0, 200) - ylims!(ax, 0, 1) - fig -end - -# ics3 = (q = ics.q[:, 1], p = ics.p[:, 1]) -# -# time_series2 = iterate(sympnet_nn, ics3; n_points = n_time_steps) -# -# function plot_solution3(time_step; theme = :light, framerate = 50) -# textcolor = theme == :dark ? :white : :black -# fig = Figure(size = (1000, 500), figure_padding = (5,50,5,5), fontsize = 24) -# ax = Axis(fig[1, 1], backgroundcolor = :transparent, -# bottomspinecolor = textcolor, -# topspinecolor = textcolor, -# leftspinecolor = textcolor, -# rightspinecolor = textcolor, -# xtickcolor = textcolor, -# ytickcolor = textcolor, -# xticklabelcolor = textcolor, -# yticklabelcolor = textcolor, -# xlabel=L"\omega", -# ylabel=L"q", -# xlabelcolor = textcolor, -# ylabelcolor = textcolor) -# lines!(ax, sol.s.q[time_step], label = rich("FOM + Implicit Midpoint"; color = textcolor), color = mblue) -# time_series = iterate(sympnet_nn, ics3; n_points = time_step) -# # prediction = (q = time_series.q[:, end], p = time_series.p[:, end]) -# prediction = (q = time_series2.q[:, time_step], p = time_series2.p[:, time_step]) -# prediction_big = decoder(sae_nn_cpu)(prediction) -# -# lines!(ax, prediction_big.q; label = rich("SAE + SympNet"; color = textcolor), color = mpurple) -# axislegend(ax; position = :rt) -# xlims!(ax, 0, 200) -# fig -# end - -# todo -#time_steps = 1:5 # axes(time_series.q, 2) -time_steps = 1:500 # axes(time_series.q, 2) - -for time_step in time_steps - fig1 = plot_solution(time_step) - save(sae_dir * "/sae-midpoint-$(string(time_step, pad = 3)).pdf", fig1) - - fig2 = plot_solution2(time_step) - save(sae_dir * "/sae-transformer-$(string(time_step, pad = 3)).pdf", fig2) - -# fig3 = plot_solution3(time_step) -# save(sae_dir * "/sae-sympnet-$(string(time_step, pad = 3)).pdf", fig3) -end diff --git a/scripts/zygote.jl b/scripts/zygote.jl deleted file mode 100644 index cde81e962..000000000 --- a/scripts/zygote.jl +++ /dev/null @@ -1,20 +0,0 @@ -using Zygote, Printf, LinearAlgebra - -const number_data_points = 1000 - -const data_input = [[i] for i in 1:number_data_points] - -function_to_be_differentiated(input, A) = norm(A*input) - -function gradient_eval(data, num, A = rand(100000,1)) - input = data[num] - @printf "First one: " - @time Zygote.gradient(A -> function_to_be_differentiated(input, A), A)[1] - @printf "Second one:" - @time Zygote.gradient(A -> function_to_be_differentiated(data[num], A), A)[1] - @printf "\n" -end - -for i in 1:5 - gradient_eval(data_input, Int(ceil(rand()*number_data_points))) -end From e6f5decdc4e444adc88d19d22970abdd71df9855 Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Mon, 17 Aug 2026 23:19:49 +0900 Subject: [PATCH 08/12] Repair the LaTeX book after the chapters moved out The PDF workflow runs on every pull request and this branch breaks it twice, in ways none of the three gates the branch was checked against would notice: `Pkg.test()`, `check_references.jl` and `docs/make.jl` all leave the book alone. `copy_png_files` ran `find` over `build/manifolds` and `build/optimizers/manifold_related`. Documenter only creates a `build/` subdirectory for a page tree that still has pages, so both went away with the chapters, and `find` exits non-zero on a missing root -- which aborts the recipe and fails the "Some sed magic" step. The two dead roots are gone and the two that remain are guarded, so the next chapter to move out does not break this again. The title page pulled in `parallel_transport_naked.png`, which was generated by a `@example` block in the `parallel_transport.md` that moved upstream. `*.png` is gitignored, so nothing produces the file and the title page came up with a missing graphic. `tikz/tangent_vector_light.png` is built by `make latex -C docs/src/tikz` from a source that is still here and shows the same subject; swap it in if you would rather have different cover art. Six entries in `adjust_image_size.jl` resized figures from those same deleted pages. They matched nothing, which is silent -- `adjust_image_size` no-ops when the pattern is absent -- so they are removed rather than repointed. Co-Authored-By: Claude Opus 5 (1M context) --- docs/Makefile | 12 ++++++++---- docs/src/assets/preamble.tex | 7 ++++++- docs/utils/adjust_image_size.jl | 9 +++------ 3 files changed, 17 insertions(+), 11 deletions(-) diff --git a/docs/Makefile b/docs/Makefile index 4547a350e..de55b376b 100644 --- a/docs/Makefile +++ b/docs/Makefile @@ -121,11 +121,15 @@ remove_numbering_for_some_chapters: sed -i'' -e 's/\\section{Chapter Summary}/\\section*{Chapter Summary}/g' build/G*.tex; sed -i'' -e 's/\\section{References}/\\section*{References}/g' build/G*.tex; +# `build/manifolds` and `build/optimizers/manifold_related` used to be listed here. Documenter only +# creates a `build/` subdirectory for a page tree that still has pages, so those two went away with +# the chapters that moved to GeometricOptimizers -- and `find` exits non-zero on a missing root, +# which aborts the recipe and takes the PDF workflow with it. The roots that remain are guarded for +# the same reason, so that the next chapter to move out does not break the build again. copy_png_files: - find build/manifolds -name \*.png -exec cp {} build \; ; - find build/optimizers/manifold_related -name \*.png -exec cp {} build \; ; - find build/tutorials -name \*.png -exec cp {} build \; - find build/reduced_order_modeling -name \*.png -exec cp {} build \; + for d in tutorials reduced_order_modeling; do \ + if [ -d "build/$$d" ]; then find "build/$$d" -name \*.png -exec cp {} build \; ; fi; \ + done docstring_indexing: sed -i'' -e 's/\\item \\hyperlinkref{\([0-9]*\)}{\\texttt{\(.*\)}}/\\item \\hyperlinkref{\1}{\\texttt{\2}}: page \\pageref{\1}/g' build/G*.tex; diff --git a/docs/src/assets/preamble.tex b/docs/src/assets/preamble.tex index fc9b97b29..e3a2d6010 100644 --- a/docs/src/assets/preamble.tex +++ b/docs/src/assets/preamble.tex @@ -29,7 +29,12 @@ {\Huge \bfseries \sffamily \@title }\\[4ex] {\Large \@author}\\[4ex] \@date\\[8ex] -\includegraphics[height = 65mm]{parallel_transport_naked.png} +% This was `parallel_transport_naked.png`, which was generated by a `@example` block in +% `optimizers/manifold_related/parallel_transport.md` -- a page that is GeometricOptimizers' +% documentation now, so nothing produces that file any more and the title page came up with a +% missing graphic. `tikz/tangent_vector_light.png` is built by `make latex -C docs/src/tikz` from a +% source that is still here, and shows the same thing: a tangent vector to a manifold. +\includegraphics[height = 65mm]{tikz/tangent_vector_light.png} \end{center}} \aliaspagestyle{title}{empty} % suppress the page number after \maketitle \makeatother diff --git a/docs/utils/adjust_image_size.jl b/docs/utils/adjust_image_size.jl index ecf59cdbb..918da0132 100644 --- a/docs/utils/adjust_image_size.jl +++ b/docs/utils/adjust_image_size.jl @@ -18,12 +18,9 @@ function adjust_image_size(path::AbstractString, size::String, lines::Union{Tupl end new_contents = adjust_image_size(raw"tikz/tangent_vector_light.png", ".5", collection_of_lines) -new_contents = adjust_image_size(raw"manifolds/sphere_with_tangent_vec_light.png", ".5", split(new_contents, "\n")) -new_contents = adjust_image_size(raw"manifolds/sphere_with_tangent_vec_and_geodesic_light.png", ".5", split(new_contents, "\n")) -new_contents = adjust_image_size(raw"optimizers/manifold_related/parallel_transport_light.png", ".5", split(new_contents, "\n")) -new_contents = adjust_image_size(raw"optimizers/manifold_related/two_vectors_light.png", ".5", split(new_contents, "\n")) -new_contents = adjust_image_size(raw"optimizers/manifold_related/retraction_comparison_light.png", ".5", split(new_contents, "\n")) -new_contents = adjust_image_size(raw"optimizers/manifold_related/retraction_discrepancy_light.png", ".5", split(new_contents, "\n")) +# The `manifolds/` and `optimizers/manifold_related/` figures used to be resized here. Both page +# trees are GeometricOptimizers' documentation now, so nothing generates those images and the calls +# matched nothing. new_contents = adjust_image_size(raw"tutorials/sympnet_training_loss_light.png", ".5", split(new_contents, "\n")) new_contents = adjust_image_size(raw"tikz/gml_venn_light.png", ".5", split(new_contents, "\n")) new_contents = adjust_image_size(raw"tikz/symplectic_autoencoder_architecture_light.png", ".65", split(new_contents, "\n")) From 7856e9ddf3c794cd6614a3663ebf6a0e00c0f9df Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Mon, 17 Aug 2026 23:20:03 +0900 Subject: [PATCH 09/12] Make the two step_size paths agree, and fix three comments MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit `Optimizer` grew two entry points for the same quantity, and they did not accept the same things. `linesearch = Static(α)` was a fixed learning rate, but `step_size = Static(α)` -- the same request through the older keyword -- fell off the end of `_optimizer_step_size` with `MethodError: no method matching _optimizer_step_size(::Static{Float64})`, leaking an internal helper's name. The carefully written `ArgumentError` explaining that a training loop has no objective for a line search to search along was only reachable from one of the two. Anything that is not a plain number now goes through that one funnel. `test/.../multi_head_attention_stiefel_optim_cache.jl` still said a blanket `using` would be ambiguous "since GeometricMachineLearning re-exports its own versions of them". That was the reason this branch removes: the shared types are one object under two names now, and `Optimizer` is the only name left that resolves to two different things. The note on the phantom `Symplectic*` exports pointed at `test/exports.jl`, which does not exist here -- it is GeometricOptimizers'. Ten exported names are still undefined in GML, which is C10, so the note now says which package the file is in and what it would take to close the class here. The GPU snippet in the symplectic autoencoder tutorial paired a `Float32` network with `AdamOptimizerWithDecay(integrator_train_epochs)`, which is `Float64` now that the type is positional rather than taken from `η₁`. `OptimizerCache` rejects that pairing, so the snippet could not have run as printed. Co-Authored-By: Claude Opus 5 (1M context) --- docs/src/tutorials/symplectic_autoencoder.md | 2 +- src/GeometricMachineLearning.jl | 4 +++- src/optimizers/optimizer.jl | 6 +++++- .../multi_head_attention_stiefel_optim_cache.jl | 6 ++++-- 4 files changed, 13 insertions(+), 5 deletions(-) diff --git a/docs/src/tutorials/symplectic_autoencoder.md b/docs/src/tutorials/symplectic_autoencoder.md index 5bcb11e7c..1777f1466 100644 --- a/docs/src/tutorials/symplectic_autoencoder.md +++ b/docs/src/tutorials/symplectic_autoencoder.md @@ -134,7 +134,7 @@ dl = DataLoader(dl_cpu, backend, Float32) sae_nn_gpu = NeuralNetwork(sae_arch, CUDADevice(), Float32) -o = Optimizer(sae_nn_gpu; AdamOptimizerWithDecay(integrator_train_epochs)...) +o = Optimizer(sae_nn_gpu; AdamOptimizerWithDecay(integrator_train_epochs, Float32)...) # train the network o(sae_nn_gpu, dl, Batch(batch_size), n_epochs) diff --git a/src/GeometricMachineLearning.jl b/src/GeometricMachineLearning.jl index 445598608..22a5ac5aa 100644 --- a/src/GeometricMachineLearning.jl +++ b/src/GeometricMachineLearning.jl @@ -120,7 +120,9 @@ export StiefelProjection export PoissonTensor # `SymplecticLieAlgMatrix`, `SymplecticLieAlgHorMatrix` and `SymplecticProjection` used to be # exported here. Nothing has defined them for as long as the git history goes back, so the exports -# were silent `UndefVarError`s waiting for a caller; see `test/exports.jl`. +# were silent `UndefVarError`s waiting for a caller. GML has no test that would have caught them — +# ten exported names are still undefined, which is issue C10; `GeometricOptimizers`' own +# `test/exports.jl` is the one-assertion-over-`names` shape that closes this class. include("kernels/assign_q_and_p.jl") include("kernels/tensor_mat_mul.jl") diff --git a/src/optimizers/optimizer.jl b/src/optimizers/optimizer.jl index e26831cca..93b766b48 100644 --- a/src/optimizers/optimizer.jl +++ b/src/optimizers/optimizer.jl @@ -116,7 +116,11 @@ _step_size(ls::DecayingStatic, t::Int) = Float64(GeometricOptimizers.step_size(l _current_step_size(opt::Optimizer, t::Int) = _step_size(opt.step_size, t) _optimizer_step_size(η::Real) = Float64(η) -_optimizer_step_size(ls::DecayingStatic) = ls +# Anything that is not a plain number goes through the same funnel as the `linesearch` keyword below, +# so that the two entry points accept the same things: `step_size = Static(α)` is a fixed learning +# rate on both, and anything else reports the `ArgumentError` that explains why a real line search +# has nothing to search along here, instead of a `MethodError` naming this helper. +_optimizer_step_size(ls) = _step_size_from_linesearch(ls) function Optimizer(method::GeometricOptimizers.OptimizerMethod, nn::NeuralNetwork; retraction = GeometricOptimizers.cayley, diff --git a/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl b/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl index 0b2bc8224..7ba9c922a 100644 --- a/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl +++ b/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl @@ -1,6 +1,8 @@ using GeometricMachineLearning, Test -# qualified access only: a blanket `using` would make `StiefelManifold` and friends ambiguous, -# since GeometricMachineLearning re-exports its own versions of them +# `StiefelManifold` and the other shared types are one object reached by two names now, so a blanket +# `using GeometricOptimizers` alongside GML would no longer be ambiguous on them -- only `Optimizer` +# still resolves to two different things (issue C1). The qualified form stays because it says which +# package owns what. import GeometricOptimizers using GeometricOptimizers: GradientCache, MomentumCache, AdamCache import Random, LinearAlgebra From d5a24173d0987b4a70e7da60aa44334641513c6b Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Wed, 19 Aug 2026 09:31:39 +0900 Subject: [PATCH 10/12] Seed the convergence tests per invocation, not per file MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit `svd_optim.jl` failed on Julia 1.10 and `sae_error_lower_than_psd_error.jl` on 1.12. Neither was a numerical regression: given the same starting point the new optimizer stack agrees with the old to 13 significant digits. Each file seeded once at the top and then called its helper twice, so the second call started from whatever RNG state the first happened to leave behind. GeometricOptimizers 0.4 builds one more `GlobalSection` per manifold parameter than 0.2 did -- GML's `StiefelManifold` *is* its type now, so its generic manifold machinery engages where `go_bridges.jl` used to -- and `global_section` calls `randn!`, so every draw after the first `Optimizer` construction shifted. Both tests were passing on a thin margin: the `svd_optim.jl` gradient run went from 2% above the optimum to 21%, against a 10% tolerance. Seeding each invocation makes the starting point independent of what ran before it. The two assertions now clear by 23x and by 2.6-4.5%, stable to 13 digits across 1.10, 1.12 and 1.13 -- 1.13 had been clearing the autoencoder comparison by 0.7%, i.e. by luck. `psd_optim.jl` and `adam_with_learning_rate_decay.jl` have the same shape and get the same treatment. The latter's manifold run also goes from 32 to 128 epochs: `AdamOptimizerWithDecay(n_epochs)` fixes γ = exp(log(η₂/η₁)/n_epochs), so a 32-epoch budget collapses the learning rate to η₂ before the run has trained, and from a seeded start the loss fell by under 2% on 1.13 and *rose* on 1.12. Full `Pkg.test()` is green on 1.10, 1.12 and 1.13 (56 testsets each). This had not been checked before: CI aborts at the first failing `@safetestset`, so most of the suite had never run on 1.10 or 1.12. Co-Authored-By: Claude Opus 5 (1M context) --- CHANGELOG.md | 21 +++++++++++++++++++ .../adam_with_learning_rate_decay.jl | 10 ++++++++- .../optimizer_convergence_tests/psd_optim.jl | 7 +++++++ .../optimizer_convergence_tests/svd_optim.jl | 7 +++++++ test/sae_error_lower_than_psd_error.jl | 7 ++++++- 5 files changed, 50 insertions(+), 2 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index 0b29f8e02..4577285d3 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -450,6 +450,27 @@ continuation lines, and reading it misses them. pins the step size by asserting that `step_size = 0` leaves the loss — which `train!` recomputes over the whole data set after every step — identical at every step. +- **Four convergence tests seed per invocation instead of once per file.** `svd_optim.jl` and + `sae_error_lower_than_psd_error.jl` failed on Julia 1.10 and 1.12 respectively, and neither was a + numerical regression: given the same starting point the new optimizer stack agrees with the old to + 13 significant digits. Each file seeded once at the top and then called its helper *twice*, so the + second call started from whatever RNG state the first happened to leave behind — and GO 0.4 builds + one more `GlobalSection` per manifold parameter than 0.2 did, because GML's `StiefelManifold` *is* + its type now and its generic manifold machinery engages where `go_bridges.jl` used to. Every draw + after the first `Optimizer` construction shifted, and both tests were passing on a thin margin: + the `svd_optim.jl` gradient run went from 2% above the optimum to 21%, against a 10% tolerance. + + Seeding each invocation makes the starting point independent of what ran before it. The two + assertions clear by 23× and by 2.6–4.5%, and are now stable to 13 digits across 1.10, 1.12 and + 1.13 — 1.13 had been passing the autoencoder comparison by 0.7%, i.e. by luck. `psd_optim.jl` and + `adam_with_learning_rate_decay.jl` have the same shape and get the same treatment; the latter's + manifold run also goes from 32 to 128 epochs, because `AdamOptimizerWithDecay(n_epochs)` fixes + `γ = exp(log(η₂/η₁)/n_epochs)` and a 32-epoch budget collapses the learning rate to `η₂` before + the run has trained — the loss fell by under 2% on 1.13 and *rose* on 1.12. The unused + `tol = .35` keyword of `sae_error_lower_than_psd_error.jl`'s `test_accuracy` is gone; the + same-named helpers in `psd_architecture_tests.jl` and `symplectic_autoencoder_tests.jl` do use + theirs and keep it. + ### Added - **`test/runtests.jl` emits seven `@info` markers**, one per testset group, so that a long job can diff --git a/test/optimizers/optimizer_convergence_tests/adam_with_learning_rate_decay.jl b/test/optimizers/optimizer_convergence_tests/adam_with_learning_rate_decay.jl index 600a4dd73..b9b53cb99 100644 --- a/test/optimizers/optimizer_convergence_tests/adam_with_learning_rate_decay.jl +++ b/test/optimizers/optimizer_convergence_tests/adam_with_learning_rate_decay.jl @@ -17,6 +17,9 @@ end # tests checks if Adam with decay achieves a lower loss value than regular Adam and the two converge reasonably well function train_network(; n_epochs=2048) + # Seeded per call, as in `svd_optim.jl`: three of these run off one top-level seed, so + # without it each one starts from whatever RNG state its predecessor left behind. + Random.seed!(123) nn₁ = setup_network(dl) nn₂ = setup_network(dl) @@ -47,7 +50,12 @@ cache explicitly, and without the routing every weight -- the `StiefelManifold` through to the Euclidean state, whose zero element is a `StiefelLieAlgHorMatrix` and not a manifold point. """ -function train_manifold_network(; n_epochs = 32) +# `n_epochs = 128` and not 32: `AdamOptimizerWithDecay(n_epochs)` fixes +# γ = exp(log(η₂/η₁)/n_epochs), so a 32-epoch budget drives the learning rate to η₂ = 1e-6 almost +# at once and the run barely trains -- from a seeded start the loss fell by under 2% on 1.13 and +# *rose* on 1.12. Over 128 epochs the decay is gentle enough that it falls by a factor of six. +function train_manifold_network(; n_epochs = 128) + Random.seed!(123) arch = Chain(StiefelLayer(1, 20), Dense(20, 20, tanh), Dense(20, 1, identity)) nn = NeuralNetwork(arch, CPU(), eltype(dl)) diff --git a/test/optimizers/optimizer_convergence_tests/psd_optim.jl b/test/optimizers/optimizer_convergence_tests/psd_optim.jl index e9ace1187..b58d66f2c 100644 --- a/test/optimizers/optimizer_convergence_tests/psd_optim.jl +++ b/test/optimizers/optimizer_convergence_tests/psd_optim.jl @@ -27,6 +27,13 @@ A = [ 0.06476993260924702 0.8369280855305259 0.6245358125914054 0.140729967064 This tests if the optimizers can find the optimal PSD solution. """ function svd_test(A, n, train_steps=1000, tol=1e-1; retraction=cayley) + # Seeded here and not once at the top of the file, for the reason given at the same place in + # `svd_optim.jl`: `svd_test` is called twice, and with a single top-level seed the second call + # starts from whatever RNG state the first one happened to leave behind, which makes the + # convergence assertions below turn on how much randomness the optimizer consumes. That is what + # broke the `StiefelLayer` version of this test on 1.10; this file has the same shape and was one + # `GlobalSection` away from the same failure. + Random.seed!(1234) N2 = size(A,1) @assert iseven(N2) N = N2÷2 diff --git a/test/optimizers/optimizer_convergence_tests/svd_optim.jl b/test/optimizers/optimizer_convergence_tests/svd_optim.jl index 72061c72b..0ea4705a1 100644 --- a/test/optimizers/optimizer_convergence_tests/svd_optim.jl +++ b/test/optimizers/optimizer_convergence_tests/svd_optim.jl @@ -26,6 +26,13 @@ A = [ 0.06476993260924702 0.8369280855305259 0.6245358125914054 0.140729967064 Random.seed!(1234) function svd_test(A, n, train_steps=1000, tol=1e-1; retraction=cayley) + # Seeded here and not once at the top of the file. `svd_test` is called twice, and with a single + # top-level seed the second call starts from whatever RNG state the first one happened to leave + # behind -- so the convergence assertions below turned on how much randomness the optimizer + # consumes rather than on whether the optimizers converge. GeometricOptimizers 0.4 builds one + # more `GlobalSection` per manifold parameter than 0.2 did, and that was enough to move the + # second pass from 2% above the optimum to 21%, against a 10% tolerance. + Random.seed!(1234) N = size(A,1) U, Σ, Vt = svd(A) U_result = U[:, 1:n] diff --git a/test/sae_error_lower_than_psd_error.jl b/test/sae_error_lower_than_psd_error.jl index 64ea81686..993de773d 100644 --- a/test/sae_error_lower_than_psd_error.jl +++ b/test/sae_error_lower_than_psd_error.jl @@ -4,7 +4,12 @@ import Random Random.seed!(123) -function test_accuracy(N::Integer, n::Integer; tol::Real = .35, n_epochs::Integer = 100) +function test_accuracy(N::Integer, n::Integer; n_epochs::Integer = 100) + # Seeded per call, for the reason given in + # `optimizers/optimizer_convergence_tests/svd_optim.jl`: this is called twice, and the second + # call inheriting the first call's RNG state made the comparison below depend on how much + # randomness the optimizer consumes instead of on whether the autoencoder beats PSD. + Random.seed!(123) dl = DataLoader(rand(N, 10 * N); autoencoder = true) psd_nn = NeuralNetwork(PSDArch(N, n)) From b00e62267b87c7ba6c6934f555a7b2311aa46bbd Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Wed, 19 Aug 2026 09:58:58 +0900 Subject: [PATCH 11/12] Correct the mechanism in the previous entry MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The previous commit blamed the shifted random stream on the optimizer state building "one more `GlobalSection` per manifold parameter". It does not: the `global_section` calls are identical on both sides, four per `Optimizer` over two Stiefel weights. The extra randomness is `_similar`. `GeometricOptimizers._similar(a::Manifold)` is `rand(manifold_constructor(a){T}, size(a)...)` -- a fresh random point on the manifold -- because upstream makes `Base.similar(::Manifold)` an error on the grounds that uninitialised storage is not a manifold point. `GradientState` allocates its `x̄` slot with it. GML's `StiefelManifold` is GeometricOptimizers' type now, so that method applies; on `main` the call fell through to `similar(a)` and GML's own `Base.similar(::StiefelManifold)`, which allocated uninitialised storage and drew nothing. Six batches of normals per `Optimizer` where there were four. The wrong claim came from a bad measurement: counting draws by locating the next scalar `randn()` in a reference list, which assumes an array fill consumes the stream the way scalar draws do. It does not, so those numbers were meaningless. Counting `randn!`/`randn` calls and reading the Xoshiro state directly is what settled it. Nothing about the fix changes -- the tests are unchanged by this commit, and the reason they needed seeding stands. Co-Authored-By: Claude Opus 5 (1M context) --- CHANGELOG.md | 17 ++++++++++++----- 1 file changed, 12 insertions(+), 5 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index 4577285d3..cddbdb05d 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -454,11 +454,18 @@ continuation lines, and reading it misses them. `sae_error_lower_than_psd_error.jl` failed on Julia 1.10 and 1.12 respectively, and neither was a numerical regression: given the same starting point the new optimizer stack agrees with the old to 13 significant digits. Each file seeded once at the top and then called its helper *twice*, so the - second call started from whatever RNG state the first happened to leave behind — and GO 0.4 builds - one more `GlobalSection` per manifold parameter than 0.2 did, because GML's `StiefelManifold` *is* - its type now and its generic manifold machinery engages where `go_bridges.jl` used to. Every draw - after the first `Optimizer` construction shifted, and both tests were passing on a thin margin: - the `svd_optim.jl` gradient run went from 2% above the optimum to 21%, against a 10% tolerance. + second call started from whatever RNG state the first happened to leave behind — and `Optimizer` + now draws more randomness than it did. `GeometricOptimizers._similar` of a manifold parameter is + `rand(manifold_constructor(a){T}, size(a)...)`, a fresh random point on the manifold, because + upstream makes `Base.similar(::Manifold)` an error on the grounds that uninitialised storage is not + a manifold point; `GradientState` allocates its `x̄` slot with it. GML's `StiefelManifold` *is* + `GeometricOptimizers`' type now, so that method applies where on `main` the call fell through to + `similar(a)` and GML's own `Base.similar(::StiefelManifold)`, which allocated uninitialised storage + and drew nothing. Constructing one optimizer over two Stiefel weights draws six batches of normals + where it drew four — the four `global_section` batches are unchanged, and each manifold parameter + adds one random manifold point. Every draw after the first `Optimizer` construction shifted, and + both tests were passing on a thin margin: the `svd_optim.jl` gradient run went from 2% above the + optimum to 21%, against a 10% tolerance. Seeding each invocation makes the starting point independent of what ran before it. The two assertions clear by 23× and by 2.6–4.5%, and are now stable to 13 digits across 1.10, 1.12 and From 9b8007e22624a8cee0f97635f55c15ecc6b2bbd8 Mon Sep 17 00:00:00 2001 From: Michael Kraus Date: Wed, 19 Aug 2026 14:19:18 +0900 Subject: [PATCH 12/12] Seed the step-size testset, which was a 1.7% flake `Julia 1.12 - windows` failed on b00e6226, a commit that touched nothing but CHANGELOG.md. The failure was `test/training_parameters.jl:54`, `!all(loss_moving .== loss_moving[1])`. The file's comment claimed the assertion needed no seed. That holds for the `step_size = 0` half -- no parameter can move, so every loss entry is the same number -- and not for the `step_size = 1e-2` half, which this file added along with the rest of the testset. `tra_ps_data` contains an all-zero trajectory, and a draw that takes only zero samples for all five runs produces a zero gradient every time and a loss array that never moves. Over 60 seeds that is one initialisation in sixty, identically on 1.10 and 1.12, so it is the initialisation and not the Julia version. Seeded at 123, where the loss spreads by 0.16 on 1.10, 1.12 and 1.13 alike -- the failure mode is a spread of exactly zero, so this is nowhere near the edge. The comment now says which half needs the seed and why. Full `Pkg.test()` green on 1.10 and 1.12, 56 testsets each. Co-Authored-By: Claude Opus 5 (1M context) --- CHANGELOG.md | 9 +++++++++ test/training_parameters.jl | 12 ++++++++++-- 2 files changed, 19 insertions(+), 2 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index cddbdb05d..2e8a23c24 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -478,6 +478,15 @@ continuation lines, and reading it misses them. same-named helpers in `psd_architecture_tests.jl` and `symplectic_autoencoder_tests.jl` do use theirs and keep it. +- **`test/training_parameters.jl` seeds its second testset.** Its `step_size = 1e-2` half asserts + `!all(loss_moving .== loss_moving[1])`, and the file's comment claimed the assertion needed no + seed. That is true of the `step_size = 0` half and false of this one: `tra_ps_data` contains an + all-zero trajectory, and a draw that takes only zero samples for all five runs gives a zero + gradient every time and a loss array that never moves. Measured over 60 seeds it happens for one + initialisation in sixty, on 1.10 and 1.12 alike — and it duly took out `Julia 1.12 - windows` on a + commit that changed nothing but this file's neighbours in the CHANGELOG. Seeded at 123, where the + loss spreads by 0.16 on all three versions. + ### Added - **`test/runtests.jl` emits seven `@info` markers**, one per testset group, so that a long job can diff --git a/test/training_parameters.jl b/test/training_parameters.jl index c12d59848..829b7ff18 100644 --- a/test/training_parameters.jl +++ b/test/training_parameters.jl @@ -1,6 +1,7 @@ using GeometricMachineLearning using GeometricMachineLearning: nruns, opt, method, batchsize using Test +import Random include("data/data_generation.jl") @@ -36,9 +37,16 @@ end # silent rather than an error — hence a test that observes whether the network moves. # # `train!` recomputes the loss over the *whole* data set after every step, so the value it stores -# does not depend on which batch was drawn. That gives an assertion needing no seed: at -# `step_size = 0` no parameter can move, hence every entry of the loss array is the same number. +# does not depend on which batch was drawn. That makes the `step_size = 0` half seed-independent: no +# parameter can move, hence every entry of the loss array is the same number. +# +# The `step_size = 1e-2` half is *not* seed-independent, which an earlier version of this comment +# claimed it was. `tra_ps_data` has an all-zero trajectory in it, and a draw that takes only zero +# samples for all five runs produces a zero gradient every time and a loss array that never moves -- +# about one initialisation in sixty, measured over 60 seeds on 1.10 and 1.12 alike. That is a 1.7% +# flake in CI, and it is why the seed is here. @testset "train! forwards a step size" begin + Random.seed!(123) m = BasicSympNet() o = GradientOptimizer() ntraining = 5