fv_tied_variations_residual#

t3toolbox.backend.sharing.fv_tied_variations_residual(variations, shared_data, rcond=None)#
def fv_tied_variations_residual(
        variations:     typ.Tuple[
            typ.Sequence[NDArray],  # tucker_variations. len=d, elm_shape=K+C+(nDi, Ni)
            typ.Sequence[NDArray],  # tt_variations.     len=d, elm_shape=K+C+(rLi, nUi, rR(i+1))
        ],
        shared_data:    'SharedFrameData',  # the frame's companion (fv_shared_frame_data)
        rcond:          typ.Optional[float] = None,  # relative clip on the group spectrum; None -> dtype eps
) -> NDArray:  # shape = K + C; relative deviation per stack element (0 == already tied)

How far a tangent’s coordinates are from the TIED tangent subspace, per stack element.

One global Frobenius ratio: ||Pi_sh(V) - V||_F / ||V||_F, with both norms taken over all d Tucker variation cores at once (sum of squares, then one square root) and the stack axes K + C kept. Only the Tucker variations can be untied – the TT variations are unrestricted – so they alone enter the norm. Zero reference with a nonzero deviation gives inf, branch-free, matching t3_sharing_residual().

This is the non-enforcing checker behind the shared geometry’s TIED-tangent precondition. It costs one tied projection, which is strictly cheaper than the retraction it guards.

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.manifold as t3m
>>> import t3toolbox.shared_geometry as sg
>>> import t3toolbox.backend.sharing as sharing
>>> np.random.seed(0)
>>> sh = (0, 0, 1)
>>> x = t3.TuckerTensorTrain.randn((6, 6, 5), (2, 2, 2), (1, 2, 2, 1)).share(sh)
>>> geom = sg.shared_manifold(sh)
>>> frame = geom.frame(x)
>>> companion = geom.shared_frame_data(frame)

A tangent produced by the shared geometry is already tied; a raw one from the base geometry is not:

>>> tied = geom.randn(frame)
>>> print(bool(sharing.fv_tied_variations_residual(tied.variations.data, companion) < 1e-12))
True
>>> raw = t3m.MANIFOLD.randn(frame)
>>> print(bool(sharing.fv_tied_variations_residual(raw.variations.data, companion) > 0.1))
True
Parameters:

  • variations (t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.typ.Sequence[NDArray], t3toolbox.backend.common.typ.Sequence[NDArray]])

  • shared_data (SharedFrameData)

  • rcond (t3toolbox.backend.common.typ.Optional[float])

Return type:

NDArray