compute_tau_tilde
=================

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

   .. code-block:: python

      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 <https://arxiv.org/abs/2603.21141>`_