left_svd

t3toolbox.backend.linalg.left_svd(G0_i_a_j, min_rank=None, max_rank=None, rtol=None, atol=None)

def left_svd(
        G0_i_a_j: NDArray, # shape=(..., ni, na, nj)
        min_rank: int = None, # 1 <= min_rank <= max_rank <= minimum(ni*na, nj)
        max_rank: int = None, # 1 <= min_rank <= max_rank <= minimum(ni*na, nj)
        rtol: float = None, # removes singular values satisfying sigma < maximum(atol, rtol*sigma1)
        atol: float = None, # removes singular values satisfying sigma < maximum(atol, rtol*sigma1)
) -> typ.Tuple[
    NDArray, # U_i_a_x, shape=(..., ni, na, r)
    NDArray, # ss_x,    shape=(.., r)
    NDArray, # Vt_x_j,  shape=(..., r, nj)
]:

Compute (truncated) singular value decomposition of 3-tensor left unfolding.

First two indices of the tensor are grouped for the SVD: G[i,a,j] = sum_x U[i,a,x] ss[x] Vt[x,j], with U orthonormal in its grouped (i,a) rows. Truncation args behave as in truncated_svd().

Examples

>>> import numpy as np
>>> import t3toolbox.backend.linalg as linalg
>>> np.random.seed(0)
>>> G = np.random.randn(4, 5, 6)                      # (ni, na, nj)
>>> U, ss, Vt = linalg.left_svd(G)
>>> print(U.shape, ss.shape, Vt.shape)               # U keeps (i,a); Vt is the 2d right factor
(4, 5, 6) (6,) (6, 6)
>>> print(np.allclose(np.einsum('iax,x,xj->iaj', U, ss, Vt), G))   # reconstructs G
True
>>> Um = U.reshape(4 * 5, -1)
>>> print(np.allclose(Um.T @ Um, np.eye(Um.shape[1])))   # left unfolding of U is orthonormal
True

Parameters:

• G0_i_a_j (NDArray)

• min_rank (int)

• max_rank (int)

• rtol (float)

• atol (float)

Return type:

t3toolbox.backend.common.typ.Tuple[NDArray, NDArray, NDArray]