TuckerTensorTrain.probe_ambient_transpose#

static t3toolbox.tucker_tensor_train.TuckerTensorTrain.probe_ambient_transpose(ztildes, ww, sum_over_probes=False)#
def probe_ambient_transpose(
        ztildes: Sequence[NDArray],  # probe residuals, len=d, elm_shape = W + C + (Ni,)
        ww:      Sequence[NDArray],  # probe vectors,   len=d, elm_shape = W + (Ni,)
        sum_over_probes: bool = False,
) -> Sequence[NDArray]:             # canonical (CP) factors, len=d, elm_shape = stack + (R, Ni)

Ambient transpose of probe(): back-project probe residuals into canonical (CP) factors.

The probe counterpart of apply_ambient_transpose() – the literal adjoint of probe as a linear map on the full tensor space (one of three probe transposes; full taxonomy in docs/transposes.md). probe returns d vectors, so the residual ztildes is d vectors; the back-projection is the rank-``d`` tensor sum_i (w0 (x) ... (x) ztildes_i (x) ... (x) w_{d-1}) (residual ztildes_i in slot i, probe vectors elsewhere), returned as CP factors. Frame-free. Realize a TuckerTensorTrain with from_canonical().

The other two probe transposes are probe_corewise_transpose (gradient w.r.t. a frame’s cores, for Adam / L-BFGS) and T3Tangent.probe_transpose (the Riemannian gradient).

sum_over_probes=False keeps the probe stack W (a W (+ C) stack of rank-d CPs); True folds W into the CP rank (one rank-d|W| CP, the ambient J^T r), cheap as CP.

See also

probe, apply_ambient_transpose, probe_corewise_transpose, from_canonical

Examples

Adjoint identity <probe_ambient_transpose(z, ww), x>_F == sum_i <z_i, x.probe(ww)_i>:

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> np.random.seed(0)
>>> x = t3.TuckerTensorTrain.randn((10, 11, 12), (5, 6, 4), (1, 2, 3, 1))
>>> ww = (np.random.randn(10), np.random.randn(11), np.random.randn(12))
>>> zt = (np.random.randn(10), np.random.randn(11), np.random.randn(12))
>>> T = t3.TuckerTensorTrain.from_canonical(t3.TuckerTensorTrain.probe_ambient_transpose(zt, ww))
>>> lhs = float(np.sum(T.to_dense() * x.to_dense()))
>>> rhs = float(sum(np.sum(z * p) for z, p in zip(zt, x.probe(ww))))
>>> print(bool(abs(lhs - rhs) < 1e-9))
True
Parameters:

  • ztildes (collections.abc.Sequence[NDArray])

  • ww (collections.abc.Sequence[NDArray])

  • sum_over_probes (bool)

Return type:

collections.abc.Sequence[NDArray]