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The Moore-Penrose pseudo-inverse A⁺ generalizes matrix inverse to non-square and rank-deficient matrices. For full-rank square matrices, A⁺ = A⁻¹. For least-squares problems, x = A⁺b minimizes ||Ax - b||₂. Pseudo-inverse is computed via SVD: A = UΣV^T → A⁺ = VΣ⁺U^T where Σ⁺ inverts non-zero singular values.
Only keep singular values above threshold, discard noisy small values.
Use optimized SVD routine for pseudo-inverse computation.
Improve pseudo-inverse via Newton iteration: A⁺ ← 2A⁺ - A⁺AA⁺.
Naive inversion of all singular values amplifies numerical noise.
void pseudo_inverse_naive(float* d_A, float* d_Ainv, int m, int n) {
float *d_U, *d_S, *d_V;
cudaMalloc(&d_U, m * m * sizeof(float));
cudaMalloc(&d_S, min(m,n) * sizeof(float));
cudaMalloc(&d_V, n * n * sizeof(float));
// Full SVD
compute_full_svd(d_A, d_U, d_S, d_V, m, n);
// Invert all singular values (dangerous for small values!)
invert_singular_values<<<...>>>(d_S, min(m,n));
// A+ = V * S+ * U^T
// ... multiple matrix multiplications
}Truncated pseudo-inverse provides numerical stability by ignoring small singular values.
void pseudo_inverse_truncated(cusolverDnHandle_t solver, cublasHandle_t blas,
float* d_A, float* d_Ainv, int m, int n, float tol) {
int min_mn = min(m, n);
float *d_U, *d_S, *d_VT;
cudaMalloc(&d_U, m * min_mn * sizeof(float));
cudaMalloc(&d_S, min_mn * sizeof(float));
cudaMalloc(&d_VT, min_mn * n * sizeof(float));
// Thin SVD (economy size)
cusolverDnSgesvd(solver, 'S', 'S', m, n, d_A, m,
d_S, d_U, m, d_VT, min_mn, ...);
// Truncated inversion - only invert σ > tol * σ_max
truncate_and_invert<<<...>>>(d_S, min_mn, tol);
// A+ = V * S+ * U^T efficiently
// Scale columns of U by S+, then multiply V^T
scale_columns<<<...>>>(d_U, d_S, m, min_mn);
cublasSgemm(blas, CUBLAS_OP_T, CUBLAS_OP_T, n, m, min_mn,
&one, d_VT, min_mn, d_U, m, &zero, d_Ainv, n);
}
__global__ void truncate_and_invert(float* S, int n, float tol) {
int i = blockIdx.x * blockDim.x + threadIdx.x;
if (i >= n) return;
float sigma_max = S[0]; // Assuming sorted descending
S[i] = (S[i] > tol * sigma_max) ? 1.0f / S[i] : 0.0f;
}| Metric | Naive | Optimized | Improvement |
|---|---|---|---|
| 2048x1024 matrix | 850ms | 420ms (thin SVD) | 2x faster |
| Memory usage | O(m² + n²) | O(mn) | ~4x less |
| Numerical accuracy (ill-cond) | 1e-2 error | 1e-6 error (truncated) | 10000x better |
Common choices: (1) tol = ε * max(m,n) * σ_max where ε is machine precision, (2) tol based on noise level in data, (3) Use numerical rank estimation. Default in NumPy: tol = max(m,n) * ε * σ_max.
Direct least squares (QR or normal equations) is faster for solving Ax≈b once. Pseudo-inverse is useful when solving multiple systems with same A, or when you need the explicit inverse operator.
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