T3Tangent.apply

t3toolbox.manifold.T3Tangent.apply(ww)

def apply(
        self,
        ww:         typ.Sequence[NDArray],  # apply vectors, len=d, elm_shape=W+(Ni,)
) -> NDArray:                               # apply(v, ww), one scalar per stack element; shape=W+K+C

Apply this tangent vector in all modes: contract the dense tangent with ww everywhere.

The all-modes special case of probe() (probing leaves one index free; this contracts them all). It is the tangent analogue of TuckerTensorTrain.apply(), and is cheaper than probing – a single left-to-right sweep, no right/central sweeps, no per-mode assembly. The result is stacked W + K + C (apply-vector stack W outer, tangent stack K, frame stack C inner); a plain scalar when there are no stacks.

See Section 6.2.2 (Algorithms 6-7) of Alger et al. (2026), “Tucker Tensor Train Taylor Series” (arXiv:2603.21141).

See also

entries, probe

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.frame_variations_format as bvf
>>> import t3toolbox.manifold as t3m
>>> x = t3.TuckerTensorTrain.randn((10, 11, 12), (5, 6, 4), (1, 2, 3, 1))
>>> frame, variations = bvf.t3_orthogonal_representations(x)
>>> v = t3m.T3Tangent(frame, variations)
>>> ww = (np.random.randn(10), np.random.randn(11), np.random.randn(12))
>>> a = v.apply(ww)
>>> a_dense = np.einsum('ijk,i,j,k->', v.to_dense(), *ww)   # dense reference
>>> print(bool(abs(float(a) - float(a_dense)) < 1e-9))
True

Parameters:

ww (t3toolbox.backend.common.typ.Sequence[NDArray])

Return type:

NDArray