t3_probe_ambient_transpose
==========================

.. py:function:: t3toolbox.backend.probing.t3_probe_ambient_transpose(ztildes, ww, sum_over_probes = False)

   .. code-block:: python

      def t3_probe_ambient_transpose(
              ztildes:    typ.Sequence[NDArray],  # probe residuals, len=d, elm_shape=W+C+(Ni,)
              ww:         typ.Sequence[NDArray],  # probe vectors,   len=d, elm_shape=W+(Ni,)
              sum_over_probes: bool = False,      # True: W folds into the CP rank
      ) -> typ.Sequence[NDArray]:  # canonical (CP) factors. len=d, ith elm_shape=stack_shape+(R, Ni)


   Ambient transpose of :py:func:`t3_probe`: back-project probe residuals into CP factors.

   The *ambient* adjoint -- the transpose of ``probe`` as a linear map on the **full tensor space**.
   Probe returns ``d`` vectors (one free mode each), so the residual ``ztildes`` is ``d`` vectors; the
   back-projection is the rank-``d`` tensor

       sum_i  w0 (x) ... (x) w_{i-1} (x) ztildes_i (x) w_{i+1} (x) ... (x) w_{d-1}

   (term ``i`` has the residual ``ztildes_i`` in slot ``i`` and the probe vectors elsewhere), whose
   natural representation is a **canonical (CP) decomposition** of rank ``d``. Frame-free. Distinct from
   the *corewise* transpose (gradient w.r.t. a frame's cores) and the *tangent* transpose (Riemannian
   gradient); see ``docs/transposes.md``. The ``apply``/``entries`` analog is the rank-1 (or rank-``|W|``)
   :py:func:`t3_apply_ambient_transpose`.

   - ``sum_over_probes=False`` (primary): ``W`` is a passthrough stacking axis -- a ``W (+ C)`` stack
     of rank-``d`` CP tensors.
   - ``sum_over_probes=True``: ``W`` folds into the CP rank -- one rank-``d|W|`` CP tensor
     ``sum_W sum_i (...)``. Cheap as CP (``O(d |W| N)``).

   Returns CP ``factors`` (factor ``k`` has the diagonal structure: rank slot ``k`` = ``ztildes_k``,
   the others = ``ww_k``), in the layout :py:func:`t3_conversions.t3_from_canonical` consumes.