T3Tangent.probe

t3toolbox.manifold.T3Tangent.probe(ww)

def probe(
        self,
        ww:         typ.Sequence[NDArray],  # probing vectors, len=d, elm_shape=W+(Ni,)
) -> typ.Sequence[NDArray]:                 # probes, len=d, elm_shape=W+K+C+(Ni,)

Probe this tangent vector: apply the single-sample least-squares Jacobian J^(s).

Contracts the tangent vector with the probing vectors ww in all-but-one index, for each index – the tangent analogue of TuckerTensorTrain.probe(). The probes are stacked W + K + C (probe stack W from ww outermost, tangent stack K next, frame stack C innermost). K is empty unless this is a tangent-stacked (K-stacked) T3Tangent, in which case J^(s) is applied to each of the K tangent vectors sharing the frame.

This is the bare J^(s) (no gauge projector Pi); for the Riemannian J = J^(s) o Pi compose a gauge projection (e.g. ManifoldGeometry.project()) yourself.

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

See also

probe_transpose

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.frame_variations_format as bvf
>>> import t3toolbox.manifold as t3m
>>> import t3toolbox.backend.probing as t3p
>>> 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(2, 10), np.random.randn(2, 11), np.random.randn(2, 12))
>>> zz = v.probe(ww)
>>> print(zz[0].shape)             # W + C + (N0,) = (2,) + () + (10,)
(2, 10)
>>> zz2 = t3p.dense_probe(ww, v.to_dense())   # dense reference
>>> print(bool(max(float(np.linalg.norm(a - b)) for a, b in zip(zz, zz2)) < 1e-9))
True

A tangent-stacked (K-stacked) tangent probes each of its K vectors, output W + K + C:

>>> vb = t3m.COREWISE.randn(frame, stack_shape=(3,))
>>> zzb = vb.probe(ww)
>>> print(zzb[0].shape)            # W + K + C + (N0,) = (2,) + (3,) + () + (10,)
(2, 3, 10)

Parameters:

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

Return type:

t3toolbox.backend.common.typ.Sequence[NDArray]