t3_apply_ambient_transpose

t3toolbox.backend.apply.t3_apply_ambient_transpose(c, ww, sum_over_probes=False)

def t3_apply_ambient_transpose(
        c:                  NDArray,                # residual, shape=W+C
        ww:                 typ.Sequence[NDArray],  # apply vectors, len=d, elm_shape=W+(Ni,)
        sum_over_probes:    bool = False,           # True: W becomes the CP rank (ambient J^T r)
) -> typ.Sequence[NDArray]:  # canonical (CP) factors. len=d, ith elm_shape=stack_shape+(R, Ni)

Ambient transpose of t3_apply(): back-project c into CP factors.

The ambient adjoint – the transpose of apply as a linear map on the full tensor space (X -> ( <X, w0^W (x) ... (x) w_{d-1}^W> )_W). Frame-free; the back-projection c * (w0 (x) ... (x) w_{d-1}) is rank-1, whose natural representation is a canonical (CP) decomposition (apply consumes one vector per mode; its adjoint emits one scaled vector per mode). This is distinct from the corewise transpose (gradient w.r.t. a base point’s cores) and the tangent transpose (Riemannian gradient) – see docs/transposes.md for the full taxonomy.

• sum_over_probes=False (primary): W is a passthrough stacking axis – a W (+ C) stack of rank-1 CP tensors (CP rank R=1).

• sum_over_probes=True: W becomes the CP rank – one rank-|W| CP tensor sum_W c_W * (w0^W (x) ...) (the ambient J^T r). Cheap as CP (O(d |W| N), the shared rank index stays implicit); the |W|^2 cost of a dense Tucker tensor train is incurred only if you convert with t3_conversions.t3_from_canonical.

Returns the CP factors (c folded into the first), in the layout t3_conversions.t3_from_canonical consumes.

Parameters:

• c (NDArray)

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

• sum_over_probes (bool)

Return type:

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