compute_tau_tilde

t3toolbox.backend.probing.compute_tau_tilde(deta_tildes, left_tt_cores, xis, mus)

def compute_tau_tilde(
        deta_tildes:        typ.Union[typ.Sequence[NDArray], NDArray],  # len=d, elm_shape=(...,nUi)
        left_tt_cores:      typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(rLi,nUi,rL(i+1))
        xis:                typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(...,nUi)
        mus:                typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(...,rLi)
) -> typ.Union[typ.Sequence[NDArray], NDArray]: # len=d, elm_shape=(...,rLi)

Adjoint-var-rightward edge variables tau_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])

• left_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])

• mus (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]