TuckerTensorTrain.is_left_orthogonal

t3toolbox.tucker_tensor_train.TuckerTensorTrain.is_left_orthogonal(atol=1e-09)

def is_left_orthogonal(self, atol: float = 1e-9) -> NDArray:  # bool array, shape = stack_shape (scalar unstacked)

True (per stack element) if this T3 is in left-orthogonal form: every Tucker core down-orthogonal and every TT core except the last left-orthogonal (the last TT core is the center remainder).

Non-enforcing convenience checker (max-abs deviation from the identities <= atol; see t3_orthogonality_residual()). A t3svd() result is left-orthogonal, as is the result of rank_adjustment_sweep('left_to_right'). Per-stack-element bool array (scalar when unstacked); reduce with .all() for a single verdict.

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> np.random.seed(0)
>>> x = t3.TuckerTensorTrain.randn((6, 7, 8), (5, 6, 7), (1, 4, 3, 1))
>>> print(x.is_left_orthogonal())     # a random T3 is not in any orthogonal form
False
>>> x2, _, _ = x.t3svd()              # a t3svd result is left-orthogonal
>>> print(x2.is_left_orthogonal())
True
>>> print(x2.is_right_orthogonal())   # ...but not right-orthogonal
False

Stacked: a per-element bool array. Stack a left-orthogonal element with a non-orthogonal one:

>>> m = t3.TuckerTensorTrain.randn((5, 6, 7), (2, 2, 2), (1, 2, 2, 1))  # minimal -> t3svd keeps ranks
>>> stacked = t3.TuckerTensorTrain.stack([m.t3svd()[0], m])
>>> print(stacked.is_left_orthogonal().shape, stacked.is_left_orthogonal())
(2,) [ True False]

Parameters:

atol (float)

Return type:

NDArray