fv_tied_ambient_directions
==========================

.. py:function:: t3toolbox.backend.sharing.fv_tied_ambient_directions(variations_data, shared_data, rcond = None)

   .. code-block:: python

      def fv_tied_ambient_directions(
              variations_data: typ.Tuple[
                  typ.Sequence[NDArray],  # tucker_variations. len=d, elm_shape=(K+C)+(nDi, Ni)
                  typ.Sequence[NDArray],  # tt_variations.     len=d, elm_shape=(K+C)+(rLi, nUi, rRi)
              ],
              shared_data:    'SharedFrameData',  # the frame's companion (fv_shared_frame_data)
              rcond:          float = None,         # relative clip on the group spectrum; None -> dtype eps * max dim
      ) -> typ.Tuple[NDArray, ...]:  # per NONTRIVIAL group: Udot, elm_shape=(K+C)+(n_g, N)


   Recover each group's common gauged ambient direction ``Udot`` from (tied) Tucker
   variation coordinates -- the clipped least-squares solve of
   :py:func:`fv_share_tucker_variations`, returning ``Udot`` itself instead of the
   redistributed coordinates. Exact on exactly-tied input (``V_i = S_i^T Udot``); on untied
   input it returns the least-squares fit (the tied projection's ambient direction). The
   shared retraction uses this to build the TIED doubled-rank embedding
   (``[U_g | Udot]`` per group, with the center cores as the paired blocks).