t3_probe_ambient_transpose#

t3toolbox.backend.probing.t3_probe_ambient_transpose(ztildes, ww, sum_over_probes=False)#
def t3_probe_ambient_transpose(
        ztildes:    typ.Sequence[NDArray],  # probe residuals, len=d, elm_shape=W+C+(Ni,)
        ww:         typ.Sequence[NDArray],  # probe vectors,   len=d, elm_shape=W+(Ni,)
        sum_over_probes: bool = False,      # True: W folds into the CP rank
) -> typ.Sequence[NDArray]:  # canonical (CP) factors. len=d, ith elm_shape=stack_shape+(R, Ni)

Ambient transpose of t3_probe(): back-project probe residuals into CP factors.

The ambient adjoint – the transpose of probe as a linear map on the full tensor space. Probe returns d vectors (one free mode each), so the residual ztildes is d vectors; the back-projection is the rank-d tensor

sum_i w0 (x) … (x) w_{i-1} (x) ztildes_i (x) w_{i+1} (x) … (x) w_{d-1}

(term i has the residual ztildes_i in slot i and the probe vectors elsewhere), whose natural representation is a canonical (CP) decomposition of rank d. Frame-free. Distinct from the corewise transpose (gradient w.r.t. a frame’s cores) and the tangent transpose (Riemannian gradient); see docs/transposes.md. The apply/entries analog is the rank-1 (or rank-|W|) t3_apply_ambient_transpose().

  • sum_over_probes=False (primary): W is a passthrough stacking axis – a W (+ C) stack of rank-d CP tensors.

  • sum_over_probes=True: W folds into the CP rank – one rank-d|W| CP tensor sum_W sum_i (...). Cheap as CP (O(d |W| N)).

Returns CP factors (factor k has the diagonal structure: rank slot k = ztildes_k, the others = ww_k), in the layout t3_conversions.t3_from_canonical() consumes.

Parameters:

  • ztildes (t3toolbox.backend.common.typ.Sequence[NDArray])

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

  • sum_over_probes (bool)

Return type:

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