t3toolbox.backend.linalg.truncated_svd#

t3toolbox.backend.linalg.truncated_svd(A: NDArray, min_rank: int = None, max_rank: int = None, rtol: float = None, atol: float = None, use_jax: bool = False) → t3toolbox.backend.common.typ.Tuple[NDArray, NDArray, NDArray]#

Compute (truncated) singular value decomposition of matrix.

A = U @ diag(ss) @ Vt Equality may be approximate if truncation is used.

Parameters:

  • A (NDArray) – Matrix. shape=(N, M)

  • min_rank (int) – Minimum rank for truncation. Should have 1 <= min_rank <= max_rank <= minimum(N, M).

  • min_rank – Maximum rank for truncation. Should have 1 <= min_rank <= max_rank <= minimum(N, M).

  • rtol (float) – Relative tolerance for truncation. Remove singular values satisfying sigma < maximum(atol, rtol*sigma1).

  • atol (float) – Absolute tolerance for truncation. Remove singular values satisfying sigma < maximum(atol, rtol*sigma1).

  • xnp – Linear algebra backend. Default: np (numpy)

Returns:

  • U (NDArray) – Left singular vectors. shape=(N, k). U.T @ U = identity matrix

  • ss (NDArray) – Singular values. Non-negative. shape=(k,).

  • Vt (NDArray) – Right singular vectors. shape=(k, M) Vt @ Vt.T = identity matrix

Examples

>>> import numpy as np
>>> import t3toolbox.dense as dense
>>> A = np.random.randn(55,70)
>>> U, ss, Vt = dense.truncated_svd(A)
>>> A2 = np.einsum('ix,x,xj->ij', U, ss, Vt)
>>> print(np.linalg.norm(A - A2))
1.0428742517412705e-13
>>> rank = len(ss)
>>> print(np.linalg.norm(U.T @ U - np.eye(rank)))
1.1907994177245428e-14
>>> print(np.linalg.norm(Vt @ Vt.T - np.eye(rank)))
1.1027751835566194e-14

>>> import numpy as np
>>> import t3toolbox.dense as dense
>>> A = np.random.randn(55, 70) @ np.diag(1.0 / np.arange(1,71)**2) # Create matrix with spectral decay
>>> U, ss, Vt = dense.truncated_svd(A, rtol=1e-2) # Truncated SVD with relative tolerance 1e-2
>>> A2 = np.einsum('ix,x,xj->ij', U, ss, Vt)
>>> truncated_rank = len(ss)
>>> print(truncated_rank)
10
>>> relerr_num = np.linalg.norm(A - A2, 2) # Check error in induced 2-norm
>>> relerr_den = np.linalg.norm(A, 2)
>>> print(relerr_num / relerr_den) # should be just less than rtol=1e-2
0.008530627920514714
>>> U, ss, Vt = dense.truncated_svd(A, atol=1e-2) # Truncated SVD with absolute tolerance 1e-2
>>> A2 = np.einsum('ix,x,xj->ij', U, ss, Vt)
>>> truncated_rank = len(ss)
>>> print(truncated_rank)
24
>>> err = np.linalg.norm(A - A2, 2)  # Check error in induced 2-norm
>>> print(err) # should be just less than atol=1e-2
0.00882416786402483