t3_orthogonal_representations

t3toolbox.backend.fv_conversions.t3_orthogonal_representations(x, already_left_orthogonal=False, squash_tails=True)

def t3_orthogonal_representations(
        x: typ.Union[
            typ.Tuple[
                typ.Tuple[NDArray,...], # tucker_cores
                typ.Tuple[NDArray,...], # tt_cores
            ], # ragged
            typ.Tuple[
                NDArray, # tucker_supercore
                NDArray, # tt_supercore
            ], # uniform
        ],
        already_left_orthogonal: bool = False,
        squash_tails: bool = True,
) -> typ.Union[
    typ.Tuple[
        typ.Tuple[
            typ.Tuple[NDArray,...], # up_tucker_cores
            typ.Tuple[NDArray, ...],  # down_tucker_cores
            typ.Tuple[NDArray,...], # left_tt_cores
            typ.Tuple[NDArray,...], # right_tucker_cores
        ],
        typ.Tuple[
            typ.Tuple[NDArray,...], # tucker_variations
            typ.Tuple[NDArray,...], # tt_variations
        ],
    ], # ragged
    typ.Tuple[
        typ.Tuple[
            NDArray,  # up_tucker_supercore
            NDArray,  # down_tucker_supercore
            NDArray,  # left_tt_supercore
            NDArray,  # right_tucker_supercore
        ],
        typ.Tuple[
            NDArray,  # tucker_variations_supercore
            NDArray,  # tt_variations_supercore
        ],
    ],  # uniform
]:

Construct frame-variation representations of TuckerTensorTrain with orthogonal frame.

Sweeping orthogonalization (Algorithm 11) producing the representations (45)-(46), Appendix A.3, of Alger et al. (2026), “Tucker Tensor Train Taylor Series” (arXiv:2603.21141). NOTE: the left/right sweep order here differs from Algorithm 11 (left-then-right vs the paper’s right-then-left); the resulting orthogonal representations are equivalent.

Parameters:

• x (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.typ.Tuple[NDArray, Ellipsis], t3toolbox.backend.common.typ.Tuple[NDArray, Ellipsis]], t3toolbox.backend.common.typ.Tuple[NDArray, NDArray]])

• already_left_orthogonal (bool)

• squash_tails (bool)

Return type:

t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.typ.Tuple[NDArray, Ellipsis], t3toolbox.backend.common.typ.Tuple[NDArray, Ellipsis], t3toolbox.backend.common.typ.Tuple[NDArray, Ellipsis], t3toolbox.backend.common.typ.Tuple[NDArray, Ellipsis]], t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.typ.Tuple[NDArray, Ellipsis], t3toolbox.backend.common.typ.Tuple[NDArray, Ellipsis]]], t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.typ.Tuple[NDArray, NDArray, NDArray, NDArray], t3toolbox.backend.common.typ.Tuple[NDArray, NDArray]]]