compute_sigma_tilde#

t3toolbox.backend.probing.compute_sigma_tilde(deta_tildes, right_tt_cores, xis, nus)#
def compute_sigma_tilde(
        deta_tildes:        typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(...,nUi)
        right_tt_cores:     typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(rRi,nUi,rR(i+1))
        xis:                typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(...,nUi)
        nus:                typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(...,rR(i+1))
) -> typ.Union[typ.Sequence[NDArray], NDArray]: # len=d, elm_shape=(...,rR(i+1))

Adjoint-var-leftward edge variables sigma_tilde. Used for computing the transpose of the map from a tangent vector to its probes.

See Section 6.2.3, particularly Algorithm 8, in:

Alger, N., Christierson, B., Chen, P., & Ghattas, O. (2026). “Tucker Tensor Train Taylor Series.” arXiv preprint arXiv:2603.21141. https://arxiv.org/abs/2603.21141

Parameters:

  • deta_tildes (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

  • right_tt_cores (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

  • xis (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

  • nus (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

Return type:

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