t3toolbox.tucker_tensor_train.t3svd_dense#

t3toolbox.tucker_tensor_train.t3svd_dense(T: t3toolbox.backend.common.NDArray, min_tucker_ranks: t3toolbox.backend.common.typ.Sequence[int] = None, max_tucker_ranks: t3toolbox.backend.common.typ.Sequence[int] = None, min_tt_ranks: t3toolbox.backend.common.typ.Sequence[int] = None, max_tt_ranks: t3toolbox.backend.common.typ.Sequence[int] = None, rtol: float = None, atol: float = None, use_jax: bool = False) → t3toolbox.backend.common.typ.Tuple[TuckerTensorTrain, t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.NDArray, Ellipsis], t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.NDArray, Ellipsis]]#

Compute TuckerTensorTrain and edge singular values for dense tensor.

Parameters:

  • T (NDArray) – The dense tensor. shape=(N1, …, Nd)

  • min_tucker_ranks (typ.Sequence[int]) – Minimum Tucker ranks for truncation. len=d. e.g., (3,3,3)

  • max_tucker_ranks (typ.Sequence[int]) – Maximum Tucker ranks for truncation. len=d. e.g., (5,5,5)

  • min_tt_ranks (typ.Sequence[int]) – Minimum TT-ranks for truncation. len=d+1. e.g., (1,3,3,3,1)

  • max_tt_ranks (typ.Sequence[int]) – Maximum TT-ranks for truncation. len=d+1. e.g., (1,5,5,5,1)

  • rtol (float) – Relative tolerance for truncation.

  • atol (float) – Absolute tolerance for truncation.

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

Returns:

  • TuckerTensorTrain – Tucker tensor train approxiamtion of T

  • typ.Tuple[NDArray,…] – Singular values of matricizations. len=d. elm_shape=(ni,)

  • typ.Tuple[NDArray,…] – Singular values of unfoldings. len=d+1. elm_shape=(ri,)

See also

truncated_svd, tucker_svd_dense, tt_svd_dense, t3_svd

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> T0 = np.random.randn(40, 50, 60)
>>> c0 = 1.0 / np.arange(1, 41)**2
>>> c1 = 1.0 / np.arange(1, 51)**2
>>> c2 = 1.0 / np.arange(1, 61)**2
>>> T = np.einsum('ijk,i,j,k->ijk', T0, c0, c1, c2) # Preconditioned random tensor
>>> x, ss_tucker, ss_tt = t3.t3svd_dense(T, rtol=1e-3) # Truncate T3-SVD to reduce rank
>>> print(x.uniform_structure)
((40, 50, 60), (11, 10, 12), (1, 11, 12, 1), ())
>>> T2 = x.to_dense()
>>> print(np.linalg.norm(T - T2) / np.linalg.norm(T)) # should be slightly more than rtol=1e-3
0.002920893302364434