TuckerTensorTrain.entries_ambient_transpose

static t3toolbox.tucker_tensor_train.TuckerTensorTrain.entries_ambient_transpose(c, index, shape, sum_over_probes=False)

def entries_ambient_transpose(
        c:      NDArray,            # residual, shape = W + C
        index:  NDArray,            # int, shape = (d,) + W
        shape:  Sequence[int],      # ambient dims (N0, ..., N(d-1))
        sum_over_probes: bool = False,
) -> Sequence[NDArray]:            # canonical (CP) factors, len=d, elm_shape = stack + (R, Ni)

Ambient transpose of entries(): scatter c at index into canonical (CP) factors.

The entries counterpart of apply_ambient_transpose() – identical, with the apply vectors replaced by the unit vectors e_{index_k}, so the CP factors are one-hots and the back-projection is c * e_{idx_0} (x) ... (x) e_{idx_{d-1}}. See that method (and docs/transposes.md) for the ambient vs corewise vs tangent distinction – this is the ambient one (the frame-free adjoint on the full tensor space); the gradient-for-optimizers versions are entries_corewise_transpose and T3Tangent.entries_transpose.

sum_over_probes=True makes W the CP rank (scatter-adding colliding indices – the J^T r for entry sampling); False keeps W as a stacking axis. shape supplies the ambient dims (N0, ..., N(d-1)), which (unlike apply_ambient_transpose(), where ww carries them) c and index alone do not determine. Returns CP factors for from_canonical().

See also

entries, entries_corewise_transpose, apply_ambient_transpose, from_canonical

Examples

Back-projecting a residual scatters it at the index (realize the CP factors with from_canonical()):

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> x = t3.TuckerTensorTrain.randn((10, 11, 12), (5, 6, 4), (1, 2, 3, 1))
>>> factors = t3.TuckerTensorTrain.entries_ambient_transpose(2.0, (3, 5, 7), x.shape)
>>> ETc = t3.TuckerTensorTrain.from_canonical(factors)
>>> print(bool(abs(float(ETc.to_dense()[3, 5, 7]) - 2.0) < 1e-9))     # c lands at the index
True
>>> rest = ETc.to_dense().copy(); rest[3, 5, 7] = 0.0
>>> print(bool(np.linalg.norm(rest) < 1e-9))                          # zero elsewhere
True

Parameters:

• c (NDArray)

• index (NDArray)

• shape (collections.abc.Sequence[int])

• sum_over_probes (bool)

Return type:

collections.abc.Sequence[NDArray]