feat(math): add central finite-difference gradient-check test helper

TESTING.md

@@ -5,6 +5,17 @@ its family**, not golden output. This is the reference for the `unit-tester` age
 writing **unit tests** for `numerical/`. It answers one question per algorithm: *which mathematical
 properties must a correct implementation satisfy, and how do we assert them?*
 
+## Gradient-check test helper
+
+`numerical/math/test_doubles/GradientCheck.hpp` (namespace `math::test`, `numerical.math_test_helper`):
+
+- `CentralDifferenceGradient(f, x[, h])` — central differences; `h` is a scalar, a per-component
+  vector, or omitted for the default `cbrt(eps) · max(1, |x_i|)` (`DefaultFiniteDifferenceSteps`).
+- `ExpectGradientNear(analytic, f, x[, h], tol)` — `ADD_FAILURE` per component whose absolute **and**
+  relative errors both exceed `tol`.
+
+Use it for the **M1 gradient check** of every `Gradient`/`Backward` implementation (§7, §10).
+
 ## Rationale
 
 A numerical algorithm is not validated by "it compiles and doesn't crash." Every family has a small

doc/math/MatrixOperations.md

@@ -54,6 +54,16 @@ The skew-symmetric part $\tfrac{1}{2}(M - M^\top)$ is `Symmetrize`'s companion.
 
 Operate on `math::Matrix` / `math::SquareMatrix`. Consumed by the `filters::active` Kalman family and `estimators::ExpectationMaximization`; `CongruenceTransform` pairs naturally with `Symmetrize` for covariance-positivity hygiene.
 
+## Zero-dimension Vectors
+
+Matrix and vector dimensions must be strictly positive; a zero-length parameter vector is not
+representable. Element access on an empty fixed-size array has no valid index, so allowing it would
+trade a compile-time error for silent undefined behaviour.
+
+A component without trainable parameters (for example a pooling or pure activation stage) should
+report the absence of parameters explicitly — an optional parameter vector that is empty, or an
+interface that has no parameter accessor at all — rather than returning a zero-length vector.
+
 ## References & Further Reading
 
 - Golub, G. H. & Van Loan, C. F., "Matrix Computations", 4th ed., §2 (symmetric/skew decomposition)

numerical/math/test/CMakeLists.txt

@@ -13,6 +13,7 @@ numerical_link_qemu_runtime(numerical.math_test)
 target_sources(numerical.math_test PRIVATE
     TestCholeskyDecomposition.cpp
     TestComplexNumber.cpp
+    TestGradientCheck.cpp
     TestMath.cpp
     TestConsistencyMetrics.cpp
     TestCordic.cpp

numerical/math/test/TestGradientCheck.cpp

@@ -0,0 +1,61 @@
+#include "numerical/math/test_doubles/GradientCheck.hpp"
+#include <cmath>
+#include <gtest/gtest-spi.h>
+#include <gtest/gtest.h>
+#include <limits>
+
+namespace
+{
+    using Vector3 = math::Vector<float, 3>;
+
+    float Cubic(const Vector3& x)
+    {
+        return x.at(0, 0) * x.at(0, 0) * x.at(0, 0) + 2.0f * x.at(1, 0) * x.at(1, 0) + 3.0f * x.at(2, 0);
+    }
+
+    Vector3 CubicGradient(const Vector3& x)
+    {
+        return Vector3{ { 3.0f * x.at(0, 0) * x.at(0, 0) }, { 4.0f * x.at(1, 0) }, { 3.0f } };
+    }
+
+    class TestGradientCheck
+        : public ::testing::Test
+    {
+    public:
+        const Vector3 x{ { 2.0f }, { -1.0f }, { 0.5f } };
+    };
+}
+
+TEST_F(TestGradientCheck, central_difference_matches_analytic_gradient)
+{
+    const auto numeric = math::test::CentralDifferenceGradient(Cubic, x, 1e-2f);
+    const auto analytic = CubicGradient(x);
+
+    EXPECT_NEAR(numeric.at(0, 0), analytic.at(0, 0), 1e-3f);
+    EXPECT_NEAR(numeric.at(1, 0), analytic.at(1, 0), 1e-3f);
+    EXPECT_NEAR(numeric.at(2, 0), analytic.at(2, 0), 1e-3f);
+}
+
+TEST_F(TestGradientCheck, default_step_scales_with_component_magnitude)
+{
+    const float cbrtEps = std::cbrt(std::numeric_limits<float>::epsilon());
+
+    const auto h = math::test::DefaultFiniteDifferenceSteps(Vector3{ { 0.1f }, { -100.0f }, { 1.0f } });
+
+    EXPECT_FLOAT_EQ(h.at(0, 0), cbrtEps);
+    EXPECT_FLOAT_EQ(h.at(1, 0), 100.0f * cbrtEps);
+    EXPECT_FLOAT_EQ(h.at(2, 0), cbrtEps);
+}
+
+TEST_F(TestGradientCheck, expect_gradient_near_accepts_correct_gradient)
+{
+    math::test::ExpectGradientNear(CubicGradient(x), Cubic, x, 1e-3f);
+}
+
+TEST_F(TestGradientCheck, expect_gradient_near_reports_wrong_component)
+{
+    auto wrong = CubicGradient(x);
+    wrong.at(1, 0) = 99.0f;
+
+    EXPECT_NONFATAL_FAILURE(math::test::ExpectGradientNear(wrong, Cubic, x, 1e-2f, 1e-3f), "component[1]");
+}

numerical/math/test_doubles/CMakeLists.txt

@@ -7,5 +7,6 @@ target_link_libraries(numerical.math_test_helper INTERFACE
 )
 
 target_sources(numerical.math_test_helper PRIVATE
+    GradientCheck.hpp
     MatrixTestSupport.hpp
 )

numerical/math/test_doubles/GradientCheck.hpp

@@ -0,0 +1,83 @@
+#pragma once
+
+#include "numerical/math/Matrix.hpp"
+#include <algorithm>
+#include <cmath>
+#include <gtest/gtest.h>
+#include <limits>
+
+namespace math::test
+{
+    template<typename T, std::size_t N>
+    math::Vector<T, N> DefaultFiniteDifferenceSteps(const math::Vector<T, N>& x)
+    {
+        static_assert(std::is_floating_point_v<T>, "DefaultFiniteDifferenceSteps requires a floating-point type");
+        const T cbrtEps = std::cbrt(std::numeric_limits<T>::epsilon());
+        math::Vector<T, N> h;
+        for (std::size_t i = 0; i < N; ++i)
+            h.at(i, 0) = cbrtEps * std::max(T{ 1 }, std::abs(x.at(i, 0)));
+        return h;
+    }
+
+    template<typename T, std::size_t N, typename Func>
+    math::Vector<T, N> CentralDifferenceGradient(Func f, const math::Vector<T, N>& x, const math::Vector<T, N>& h)
+    {
+        static_assert(std::is_floating_point_v<T>, "CentralDifferenceGradient requires a floating-point type");
+        math::Vector<T, N> gradient;
+        for (std::size_t i = 0; i < N; ++i)
+        {
+            math::Vector<T, N> xPlus = x;
+            math::Vector<T, N> xMinus = x;
+            xPlus.at(i, 0) += h.at(i, 0);
+            xMinus.at(i, 0) -= h.at(i, 0);
+            gradient.at(i, 0) = (f(xPlus) - f(xMinus)) / (xPlus.at(i, 0) - xMinus.at(i, 0));
+        }
+        return gradient;
+    }
+
+    template<typename T, std::size_t N, typename Func>
+    math::Vector<T, N> CentralDifferenceGradient(Func f, const math::Vector<T, N>& x, T h)
+    {
+        math::Vector<T, N> steps;
+        for (std::size_t i = 0; i < N; ++i)
+            steps.at(i, 0) = h;
+        return CentralDifferenceGradient(f, x, steps);
+    }
+
+    template<typename T, std::size_t N, typename Func>
+    math::Vector<T, N> CentralDifferenceGradient(Func f, const math::Vector<T, N>& x)
+    {
+        return CentralDifferenceGradient(f, x, DefaultFiniteDifferenceSteps(x));
+    }
+
+    template<typename T, std::size_t N, typename Func>
+    void ExpectGradientNear(const math::Vector<T, N>& analytic, Func f, const math::Vector<T, N>& x, const math::Vector<T, N>& h, T tol)
+    {
+        const auto numeric = CentralDifferenceGradient(f, x, h);
+        for (std::size_t i = 0; i < N; ++i)
+        {
+            const T a = analytic.at(i, 0);
+            const T n = numeric.at(i, 0);
+            const T absErr = std::abs(a - n);
+            const T relErr = absErr / (std::max(std::abs(a), std::abs(n)) + std::numeric_limits<T>::min());
+            if (absErr > tol && relErr > tol)
+                ADD_FAILURE() << "component[" << i << "] analytic=" << a << " numeric=" << n
+                              << " absErr=" << absErr << " relErr=" << relErr << " tol=" << tol;
+        }
+    }
+
+    template<typename T, std::size_t N, typename Func>
+    void ExpectGradientNear(const math::Vector<T, N>& analytic, Func f, const math::Vector<T, N>& x, T h, T tol)
+    {
+        math::Vector<T, N> steps;
+        for (std::size_t i = 0; i < N; ++i)
+            steps.at(i, 0) = h;
+        ExpectGradientNear(analytic, f, x, steps, tol);
+    }
+
+    template<typename T, std::size_t N, typename Func>
+    void ExpectGradientNear(const math::Vector<T, N>& analytic, Func f, const math::Vector<T, N>& x, T tol)
+    {
+        ExpectGradientNear(analytic, f, x, DefaultFiniteDifferenceSteps(x), tol);
+    }
+}