412 lines
17 KiB
C++
412 lines
17 KiB
C++
#pragma once
|
||
#include "Matrix.hpp"
|
||
|
||
/**
|
||
* @brief library that uses Matrix.hpp and performs SVD on a matrix
|
||
*
|
||
* @note Fully templated: SVD works for ANY Matrix<R, C> with R, C in
|
||
* 1..255 (the uint8_t range of Matrix). There is no 5×5 limit.
|
||
*
|
||
* @note EMBEDDED CONSTRAINT — no heap. All working storage is stack
|
||
* allocated as templated Matrix<N,N> buffers where
|
||
* N = max(R, C). Peak stack usage per SVD call is
|
||
* ≈ 11·N² floats (≈ 44·N² bytes):
|
||
* N = 5 → ~1.1 KB
|
||
* N = 10 → ~4.4 KB
|
||
* N = 20 → ~18 KB
|
||
* N = 50 → ~110 KB
|
||
* N = 100 → ~440 KB
|
||
* N = 255 → ~2.9 MB
|
||
* Instantiate only the sizes that fit your call-stack budget.
|
||
*/
|
||
namespace SVD {
|
||
/**
|
||
* @brief Compute the Singular Value Decomposition (SVD) of this matrix.
|
||
*
|
||
* Decomposes A into U × Σ × Vᵀ where:
|
||
* - U is an m×k orthogonal matrix (left singular vectors)
|
||
* - Σ is a k×k diagonal matrix with non-negative singular values
|
||
* (stored as a k×1 column vector)
|
||
* - Vᵀ is a k×n orthogonal matrix (right singular vectors, transposed)
|
||
* - k = min(m, n)
|
||
*
|
||
* The decomposition satisfies: A ≈ U × diag(Σ) × Vᵀ
|
||
* Singular values are returned in descending order.
|
||
*
|
||
* Output storage conventions:
|
||
* - U: Matrix<rows, columns> — first k columns are meaningful
|
||
* (rows k..columns−1 are zero in the wide case)
|
||
* - sigma: Matrix<columns, 1> — first k entries are the singular
|
||
* values; entries beyond k (wide matrices only) are zero
|
||
* - Vt: Matrix<columns, columns> — first k rows are meaningful
|
||
* (zero-padded in the tall case)
|
||
*
|
||
* For wide matrices (rows < columns) the SVD is computed on Aᵀ and the
|
||
* factors are swapped back.
|
||
*
|
||
* @tparam rows Number of rows in A (1..255)
|
||
* @tparam columns Number of columns in A (1..255)
|
||
* @param matrixToDecompose Input: the matrix A
|
||
* @param U Output: left singular vectors (rows×columns matrix)
|
||
* @param sigma Output: singular values (columns×1 vector, sorted descending)
|
||
* @param Vt Output: right singular vectors transposed (columns×columns)
|
||
*
|
||
* @note This implementation uses Householder bidiagonalization followed
|
||
* by block reduction: 2×2 blocks via closed form, larger blocks
|
||
* via cyclic Jacobi eigen-decomposition of BᵀB with residual
|
||
* singular values σᵢ = ‖B·vᵢ‖ (see docs/svd-refactor.md).
|
||
*/
|
||
template <uint8_t rows, uint8_t columns>
|
||
void SVD(Matrix<rows, columns> &matrixToDecompose, Matrix<rows, columns> &U,
|
||
Matrix<columns, 1> &sigma, Matrix<columns, columns> &Vt);
|
||
|
||
// ========================================================================
|
||
// SVD Building Block Functions (for unit testing)
|
||
//
|
||
// Templated on the working-buffer size N. All block operations work on
|
||
// N×N matrices with runtime bounds (m, n, p, blockSize, ...) — the
|
||
// regions beyond the bounds are zero-padded working space.
|
||
//
|
||
// N is deduced from the Matrix arguments at the call site, e.g.
|
||
// Matrix<8, 8> W, QL, QR;
|
||
// SVD::Bidiagonalize(W, 6, 8, 6, QL, QR); // N = 8 deduced
|
||
// ========================================================================
|
||
|
||
/**
|
||
* @brief Compute a Householder reflector vector.
|
||
*
|
||
* Given input vector x, computes normalized v and scalar alpha such that:
|
||
* (I - 2·v·vᵀ) · x = [alpha, 0, 0, ...]ᵀ
|
||
*
|
||
* @param x Input vector (up to len elements)
|
||
* @param len Number of valid elements in x
|
||
* @param v Output: normalized Householder vector (length ≥ len)
|
||
* @param alpha Output: the resulting first element after reflection
|
||
* @return The norm of the input vector x
|
||
*/
|
||
static float ComputeHouseholder(const float *x, uint8_t len, float *v,
|
||
float &alpha);
|
||
|
||
/**
|
||
* @brief Apply a Householder reflection from the left.
|
||
*
|
||
* Transforms W = (I - 2·v·vᵀ) · W where v operates on rows [startRow..endRow]
|
||
* and is applied across all N columns (zero-padded columns are a no-op).
|
||
*
|
||
* @tparam N Working buffer size
|
||
* @param W Input/output: matrix to transform
|
||
* @param v Householder vector (length = endRow - startRow + 1)
|
||
* @param startRow First row index
|
||
* @param endRow Last row index
|
||
*/
|
||
template <uint8_t N>
|
||
static void ApplyHouseholderLeft(Matrix<N, N> &W, const float *v,
|
||
uint8_t startRow, uint8_t endRow);
|
||
|
||
/**
|
||
* @brief Apply a Householder reflection from the right.
|
||
*
|
||
* Transforms W = W · (I - 2·v·vᵀ) where v operates on columns
|
||
* [startCol..endCol] and is applied across all N rows (zero-padded rows
|
||
* are a no-op).
|
||
*
|
||
* @tparam N Working buffer size
|
||
* @param W Input/output: matrix to transform
|
||
* @param v Householder vector (length = endCol - startCol + 1)
|
||
* @param startCol First column index
|
||
* @param endCol Last column index
|
||
*/
|
||
template <uint8_t N>
|
||
static void ApplyHouseholderRight(Matrix<N, N> &W, const float *v,
|
||
uint8_t startCol, uint8_t endCol);
|
||
|
||
/**
|
||
* @brief Reduce a matrix to upper bidiagonal form using Householder reflections.
|
||
*
|
||
* Applies a sequence of Householder reflections to reduce the input matrix
|
||
* W (m×q, where q ≥ p, stored in N×N working space) to upper bidiagonal
|
||
* form B (p×q), accumulating the left and right transformation matrices
|
||
* in QL and QR respectively.
|
||
*
|
||
* Algorithm (Golub-Kahan bidiagonalization):
|
||
* For k = 0 to p-1:
|
||
* 1. Left HH on column k, rows k..m-1: zero out subdiagonal below B[k+1][k]
|
||
* 2. Right HH on row k, cols k+2..q-1: zero out superdiagonal above B[k][k+1]
|
||
*
|
||
* The accumulated transformations satisfy:
|
||
* QLᵀ · W_original · QR = B (upper bidiagonal)
|
||
*
|
||
* @tparam N Working buffer size (≥ m and ≥ q)
|
||
* @param W Input/output: matrix to bidiagonalize (first m×q used)
|
||
* @param m Number of rows in the working matrix
|
||
* @param q Number of columns in the working matrix (q ≥ p)
|
||
* @param p Rank = min(m, original_columns) — number of bidiagonalization steps
|
||
* @param QL Input/output: left Householder accumulation (initialized to identity)
|
||
* @param QR Input/output: right Householder accumulation (initialized to identity)
|
||
*/
|
||
template <uint8_t N>
|
||
static void Bidiagonalize(Matrix<N, N> &W, uint8_t m, uint8_t q, uint8_t p,
|
||
Matrix<N, N> &QL, Matrix<N, N> &QR);
|
||
|
||
/**
|
||
* @brief Deflate a bidiagonal matrix by zeroing negligible superdiagonals.
|
||
*
|
||
* Scans the p×p upper-bidiagonal matrix stored in W and zeros out any
|
||
* superdiagonal element W[i][i+1] whose magnitude is negligible relative
|
||
* to the local diagonal scale (|W[i][i]| + |W[i+1][i+1]|). Deflating
|
||
* splits the matrix into independent unreduced blocks that can each be
|
||
* solved separately.
|
||
*
|
||
* @tparam N Working buffer size
|
||
* @param W Input/output: bidiagonal matrix (first p×p used)
|
||
* @param p Size of the bidiagonal matrix (min(rows, columns))
|
||
* @param tol Relative deflation tolerance (e.g. 1e-8f)
|
||
*/
|
||
template <uint8_t N>
|
||
static void DeflateBidiagonal(Matrix<N, N> &W, uint8_t p, float tol);
|
||
|
||
/**
|
||
* @brief Check whether a bidiagonal matrix has fully reduced to diagonal.
|
||
*
|
||
* Returns true when every superdiagonal element of the p×p bidiagonal
|
||
* matrix in W is (numerically) zero, i.e. the diagonal entries are the
|
||
* (unsorted) singular values and no unreduced blocks remain.
|
||
*
|
||
* @tparam N Working buffer size
|
||
* @param W Input: bidiagonal matrix (first p×p used)
|
||
* @param p Size of the bidiagonal matrix (min(rows, columns))
|
||
* @param tol Numerical zero threshold multiplier
|
||
* @return true when all superdiagonal elements are ~0
|
||
*/
|
||
template <uint8_t N>
|
||
static bool BidiagonalIsDiagonal(const Matrix<N, N> &W, uint8_t p, float tol);
|
||
|
||
/**
|
||
* @brief Compute the full SVD of a 2×2 upper-bidiagonal block (pure).
|
||
*
|
||
* Decomposes B = [[a, b], [0, d]] as:
|
||
* B = Ublock · diag(sigma[0], sigma[1]) · Vblockᵀ
|
||
*
|
||
* Guarantees:
|
||
* - sigma[0] ≥ sigma[1] ≥ 0 (singular values, from eigenvalues of BᵀB)
|
||
* - Ublock and Vblock are orthogonal (columns are the left/right
|
||
* singular vectors respectively; Vblock = scipy's Vᵀᵀ)
|
||
* - Ublock · diag(sigma) · Vblockᵀ == B (within float tolerance)
|
||
*
|
||
* Math: eigenvectors of BᵀB = [[a², ab], [ab, b²+d²]] give the right
|
||
* singular vectors (v1 = normalize(ab, σ1²−a²) with a safe fallback when
|
||
* that vector is ~0; v2 = (−v1y, v1x)); left singular vectors are
|
||
* uᵢ = B·vᵢ/σᵢ with a rank-deficiency guard: when σᵢ ≈ 0 (i.e. ~1e-30),
|
||
* that U column is filled with the signed orthogonal complement of the
|
||
* other U column instead of dividing by ~0.
|
||
*
|
||
* @param a B[0][0] (first diagonal element)
|
||
* @param b B[0][1] (superdiagonal element)
|
||
* @param d B[1][1] (second diagonal element)
|
||
* @param Ublock Output: 2×2 left singular vectors (columns)
|
||
* @param Vblock Output: 2×2 right singular vectors (columns)
|
||
* @param sigma Output: singular values, sigma[0] ≥ sigma[1] ≥ 0
|
||
*/
|
||
static void SolveBidiagonalBlock2x2(float a, float b, float d,
|
||
float Ublock[2][2], float Vblock[2][2],
|
||
float sigma[2]);
|
||
|
||
/**
|
||
* @brief Cyclic Jacobi eigenvalue algorithm for a symmetric matrix (pure).
|
||
*
|
||
* Reduces symmetric n×n matrix T to (near-)diagonal form IN PLACE using
|
||
* cyclic Jacobi rotations, accumulating the eigenvectors in V.
|
||
*
|
||
* On return:
|
||
* - T's diagonal entries are the eigenvalues (off-diagonals ~0)
|
||
* - evals[i] = T[i][i], UNSORTED, SIGNED (this is a general symmetric
|
||
* eigen solver, not just for PSD matrices like T = BᵀB)
|
||
* - columns of V are the corresponding eigenvectors (T·V = V·Λ)
|
||
*
|
||
* Convergence: relative off-diagonal tolerance 1e-10, hard-capped at
|
||
* 100 sweeps.
|
||
*
|
||
* @tparam N Working buffer size (≥ n)
|
||
* @param T Input/output: symmetric matrix (first n×n used, destroyed in place)
|
||
* @param n Matrix size
|
||
* @param evals Output: eigenvalues, unsorted, length ≥ n
|
||
* @param V Output: eigenvector matrix (first n×n used), columns are eigenvectors
|
||
*/
|
||
template <uint8_t N>
|
||
static void JacobiEigenSymmetric(Matrix<N, N> &T, uint8_t n, float *evals,
|
||
Matrix<N, N> &V);
|
||
|
||
/**
|
||
* @brief Fold a block SVD's factors into the QL/QR accumulators.
|
||
*
|
||
* Given the block SVD of a bidiagonal block, B = Ublock·Σ·Vblockᵀ, the
|
||
* accumulated Householder matrices must absorb the block factors:
|
||
* QL[:, blockStart..blockStart+blockSize−1] ← QL[:, ...] · Ublock
|
||
* (rows 0..rowsQL−1)
|
||
* QR[:, blockStart..blockStart+blockSize−1] ← QR[:, ...] · Vblock
|
||
* (rows 0..rowsQR−1)
|
||
*
|
||
* rowsQL / rowsQR are the meaningful row extents of the accumulators
|
||
* (e.g. for a wide matrix W = Aᵀ, QL carries n = rows(W) meaningful
|
||
* rows while QR is read back over its first m rows).
|
||
*
|
||
* In-place update is done through temporary buffers (updating QL's block
|
||
* columns while still reading them corrupts the result).
|
||
*
|
||
* @tparam N Working buffer size
|
||
* @param blockStart First column/row index of the block in W
|
||
* @param blockSize Size of the block (2, or > 2 for the Jacobi path)
|
||
* @param Ublock Left singular-vector factor of the block (first blockSize×blockSize used)
|
||
* @param Vblock Right singular-vector factor of the block (first blockSize×blockSize used)
|
||
* @param rowsQL Number of meaningful rows of QL
|
||
* @param rowsQR Number of meaningful rows of QR
|
||
* @param QL Input/output: left transformation accumulator
|
||
* @param QR Input/output: right transformation accumulator
|
||
*/
|
||
template <uint8_t N>
|
||
static void ApplyBlockFactorsToAccumulators(uint8_t blockStart,
|
||
uint8_t blockSize,
|
||
const Matrix<N, N> &Ublock,
|
||
const Matrix<N, N> &Vblock,
|
||
uint8_t rowsQL, uint8_t rowsQR,
|
||
Matrix<N, N> &QL,
|
||
Matrix<N, N> &QR);
|
||
|
||
/**
|
||
* @brief Solve a bidiagonal block larger than 2×2 via Jacobi eigen of BᵀB.
|
||
*
|
||
* Computes the full SVD of the unreduced upper-bidiagonal block
|
||
* W[blockStart..blockStart+blockSize−1] via eigendecomposition of the
|
||
* tridiagonal T = BᵀB:
|
||
* 1. Snapshot the ORIGINAL block diagonal/superdiagonal from W
|
||
* 2. Form T = BᵀB (tridiagonal symmetric)
|
||
* 3. JacobiEigenSymmetric on T → eigenvalues (unsorted) + V
|
||
* 4. Sort eigenvalues descending, reordering V columns
|
||
* 5. Compute RESIDUAL singular values: σᵢ = ‖B_orig · vᵢ‖
|
||
* (NOT sqrt(eigenvalue) — forming BᵀB squares the condition number,
|
||
* causing float noise to swamp true tiny eigenvalues for
|
||
* rank-deficient blocks)
|
||
* 6. Re-sort σ descending, keeping V and B·v consistent
|
||
* 7. Build Ublock: uᵢ = B_orig · vᵢ / σᵢ (unit norm); for σᵢ ≈ 0,
|
||
* use Gram-Schmidt orthogonal completion against prior U columns
|
||
* 8. Fold Ublock/Vblock into QL/QR via ApplyBlockFactorsToAccumulators
|
||
* 9. Write residual norms into W's diagonal and zero the block's
|
||
* superdiagonals
|
||
*
|
||
* @tparam N Working buffer size (≥ blockSize)
|
||
* @param W Input/output: bidiagonal matrix; the block's diagonal holds
|
||
* the singular values and its superdiagonals are zeroed on return
|
||
* @param blockStart First column/row index of the block
|
||
* @param blockSize Size of the block (> 2)
|
||
* @param rowsQL Number of meaningful rows of QL
|
||
* @param rowsQR Number of meaningful rows of QR
|
||
* @param QL Input/output: left transformation accumulator
|
||
* @param QR Input/output: right transformation accumulator
|
||
*/
|
||
template <uint8_t N>
|
||
static void SolveBidiagonalBlockJacobi(Matrix<N, N> &W, uint8_t blockStart,
|
||
uint8_t blockSize, uint8_t rowsQL,
|
||
uint8_t rowsQR, Matrix<N, N> &QL,
|
||
Matrix<N, N> &QR);
|
||
|
||
/**
|
||
* @brief Extract singular values from bidiagonal matrix diagonal and sort.
|
||
*
|
||
* Extracts absolute values of diagonal elements of W as singular values,
|
||
* then sorts them in descending order while reordering columns of QL
|
||
* and QR to maintain consistency. A negative diagonal element flips the
|
||
* sign of the corresponding QL column to keep A = U·Σ·Vᵀ.
|
||
*
|
||
* @tparam N Working buffer size
|
||
* @param W Input: bidiagonal matrix (first p×p used)
|
||
* @param sigma Output: sorted singular values (N×1 column vector, only first p used)
|
||
* @param p Number of singular values (min(rows, columns))
|
||
* @param QL Input/output: left transformation matrix (modified during sort)
|
||
* @param QR Input/output: right transformation matrix (modified during sort)
|
||
*/
|
||
template <uint8_t N>
|
||
static void ExtractAndSortSingularValues(Matrix<N, N> &W, Matrix<N, 1> &sigma,
|
||
uint8_t p, Matrix<N, N> &QL,
|
||
Matrix<N, N> &QR);
|
||
|
||
/**
|
||
* @brief Assemble final U and Vt matrices from QL/QR.
|
||
*
|
||
* Computes the final left singular vectors (U) and right singular vectors
|
||
* transposed (Vt) from the accumulated Householder transformations.
|
||
*
|
||
* For non-transpose case: U = QL[:,0:p], Vt = QR[:,0:p]ᵀ
|
||
* For transpose case: U = QR[:,0:p], Vt = full QLᵀ (all n rows)
|
||
*
|
||
* @tparam N Working buffer size (≥ m and ≥ n)
|
||
* @param m Number of rows in original matrix
|
||
* @param n Number of columns in original matrix
|
||
* @param p Rank = min(m, n)
|
||
* @param transposeNeeded True if we computed SVD(Aᵀ) instead of SVD(A)
|
||
* @param QL Left Householder accumulation (N×N)
|
||
* @param QR Right Householder accumulation (N×N)
|
||
* @param U Output: left singular vectors (N×N, first m×p used)
|
||
* @param Vt Output: right singular vectors transposed (N×N, first p×n used)
|
||
*/
|
||
template <uint8_t N>
|
||
static void AssembleUAndVt(uint8_t m, uint8_t n, uint8_t p,
|
||
bool transposeNeeded, const Matrix<N, N> &QL,
|
||
const Matrix<N, N> &QR, Matrix<N, N> &U,
|
||
Matrix<N, N> &Vt);
|
||
|
||
/**
|
||
* @brief Compute a Givens rotation that zeros out y.
|
||
*
|
||
* Computes c, s such that:
|
||
* [c s] [x] = [r]
|
||
* [-s c] [y] [0]
|
||
* where r = sqrt(x² + y²).
|
||
*
|
||
* @param x First element
|
||
* @param y Second element (to be zeroed)
|
||
* @param c Output: cosine of rotation angle
|
||
* @param s Output: sine of rotation angle
|
||
*/
|
||
static void ComputeGivens(float x, float y, float &c, float &s);
|
||
|
||
/**
|
||
* @brief Apply a Givens rotation from the left to rows i and j.
|
||
*
|
||
* Applies [c s; -s c] to rows i, j of W (columns startCol..endCol).
|
||
*
|
||
* @tparam N Working buffer size
|
||
* @param W Input/output: matrix to transform
|
||
* @param i First row index
|
||
* @param j Second row index
|
||
* @param c Cosine of rotation angle
|
||
* @param s Sine of rotation angle
|
||
* @param startCol First column to transform
|
||
* @param endCol Last column to transform
|
||
*/
|
||
template <uint8_t N>
|
||
static void ApplyGivensLeft(Matrix<N, N> &W, uint8_t i, uint8_t j, float c,
|
||
float s, uint8_t startCol, uint8_t endCol);
|
||
|
||
/**
|
||
* @brief Apply a Givens rotation from the right to columns i and j.
|
||
*
|
||
* Applies [c -s; s c]ᵀ to columns i, j of W (rows startRow..endRow).
|
||
*
|
||
* @tparam N Working buffer size
|
||
* @param W Input/output: matrix to transform
|
||
* @param i First column index
|
||
* @param j Second column index
|
||
* @param c Cosine of rotation angle
|
||
* @param s Sine of rotation angle
|
||
* @param startRow First row to transform
|
||
* @param endRow Last row to transform
|
||
*/
|
||
template <uint8_t N>
|
||
static void ApplyGivensRight(Matrix<N, N> &W, uint8_t i, uint8_t j, float c,
|
||
float s, uint8_t startRow, uint8_t endRow);
|
||
} // namespace SVD
|
||
|
||
#ifndef SVD_H_
|
||
#include "SVD.cpp"
|
||
#endif
|