t3svd

t3toolbox.backend.t3_svd.t3svd(x, max_tt_ranks=None, max_tucker_ranks=None, rtol=None, atol=None, assume_orthogonal=False)

def t3svd(
        x: typ.Tuple[
            typ.Tuple[NDArray,...], # tucker_cores
            typ.Tuple[NDArray,...], # tt_cores
        ],
        max_tt_ranks:       typ.Sequence[int] = None, # len=d+1
        max_tucker_ranks:   typ.Sequence[int] = None, # len=d
        rtol: float = None,
        atol: float = None,
        assume_orthogonal: bool = False,
) -> typ.Tuple[
    typ.Tuple[
        typ.Tuple[NDArray, ...],  # new_tucker_cores
        typ.Tuple[NDArray, ...],  # new_tt_cores
    ],
    typ.Tuple[NDArray,...], # Tucker singular values, len=d
    typ.Tuple[NDArray,...], # TT singular values, len=d+1
]:

Compute (truncated) T3-SVD of TuckerTensorTrain.

Implicit T3-SVD (Algorithm 10), Appendix A.2, of Alger et al. (2026), “Tucker Tensor Train Taylor Series” (arXiv:2603.21141) – the basic algorithm, analogous to Oseledets’ TT-SVD.

Orthogonalize, then a single left-to-right truncating sweep. The result is always left-orthogonal. It is not re-tuned to minimal ranks: a hard rank cap can leave a Tucker rank / bond above its structural minimum (non-minimal), exactly as the paper’s algorithm does. To reduce to minimal ranks, follow with t3_rank_adjustment_sweep() – the output is left-orthogonal, so t3_rank_adjustment_sweep(x, 'right_to_left') minimizes it; check with TuckerTensorTrain.has_minimal_ranks. See docs/t3svd_minimal_ranks.md.

assume_orthogonal=True skips the initial orthogonalization, asserting the input is already right-orthogonal (Tucker down-orthogonal + TT right-orthogonal – the form the L->R sweep needs). Not enforced (verify with TuckerTensorTrain.is_right_orthogonal). A left-orthogonal input must be reversed by the caller (a left-orthogonal T3 reversed is right-orthogonal).

Parameters:

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

• max_tt_ranks (t3toolbox.backend.common.typ.Sequence[int])

• max_tucker_ranks (t3toolbox.backend.common.typ.Sequence[int])

• rtol (float)

• atol (float)

• assume_orthogonal (bool)

Return type:

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]]