t3_orthogonal_representations
=============================

.. py:function:: t3toolbox.backend.fv_conversions.t3_orthogonal_representations(x, already_left_orthogonal = False, squash_tails = True)

   .. code-block:: python

      def t3_orthogonal_representations(
              x: typ.Union[
                  typ.Tuple[
                      typ.Tuple[NDArray,...], # tucker_cores
                      typ.Tuple[NDArray,...], # tt_cores
                  ], # ragged
                  typ.Tuple[
                      NDArray, # tucker_supercore
                      NDArray, # tt_supercore
                  ], # uniform
              ],
              already_left_orthogonal: bool = False,
              squash_tails: bool = True,
      ) -> typ.Union[
          typ.Tuple[
              typ.Tuple[
                  typ.Tuple[NDArray,...], # up_tucker_cores
                  typ.Tuple[NDArray, ...],  # down_tucker_cores
                  typ.Tuple[NDArray,...], # left_tt_cores
                  typ.Tuple[NDArray,...], # right_tucker_cores
              ],
              typ.Tuple[
                  typ.Tuple[NDArray,...], # tucker_variations
                  typ.Tuple[NDArray,...], # tt_variations
              ],
          ], # ragged
          typ.Tuple[
              typ.Tuple[
                  NDArray,  # up_tucker_supercore
                  NDArray,  # down_tucker_supercore
                  NDArray,  # left_tt_supercore
                  NDArray,  # right_tucker_supercore
              ],
              typ.Tuple[
                  NDArray,  # tucker_variations_supercore
                  NDArray,  # tt_variations_supercore
              ],
          ],  # uniform
      ]:


   Construct frame-variation representations of TuckerTensorTrain with orthogonal frame.

   Sweeping orthogonalization (Algorithm 11) producing the representations (45)-(46), Appendix A.3,
   of Alger et al. (2026), "Tucker Tensor Train Taylor Series" (arXiv:2603.21141). NOTE: the
   left/right sweep order here differs from Algorithm 11 (left-then-right vs the paper's
   right-then-left); the resulting orthogonal representations are equivalent.