tv_oblique_gauge_projection

t3toolbox.backend.tv_operations.tv_oblique_gauge_projection(frame, variations)

def tv_oblique_gauge_projection(
        frame:      typ.Tuple[
            typ.Sequence[NDArray],  # up_tucker_cores
            typ.Sequence[NDArray],  # down_tt_cores
            typ.Sequence[NDArray],  # left_tt_cores
            typ.Sequence[NDArray],  # right_tt_cores
        ],
        variations: typ.Tuple[
            typ.Sequence[NDArray],  # tucker_variations
            typ.Sequence[NDArray],  # tt_variations
        ],
) -> typ.Tuple[
    typ.Tuple[NDArray, ...],  # gauged_tucker_variations
    typ.Tuple[NDArray, ...],  # gauged_tt_variations
]:

Project the variations onto the gauge-satisfying subspace while preserving the tangent vector.

Generalizes Holtz, Rohwedder & Schneider (2012), “On manifolds of tensors of fixed TT-rank”. The Tucker perturbation is made perpendicular to U (compensating through the down/outer cores), then the TT variations are made left-perpendicular (compensating through the right cores). Stack-aware. Ragged path only (uniform deferred).

Enforces the gauge conditions (48)-(49), Appendix A.3, of Alger et al. (2026), “Tucker Tensor Train Taylor Series” (arXiv:2603.21141).

Parameters:

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

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

Return type:

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