tv_apply_derivatives_transpose
==============================

.. py:function:: t3toolbox.backend.sampling_derivatives.tv_apply_derivatives_transpose(c, ww, pp, frame, order, sum_over_probes = False)

   .. code-block:: python

      def tv_apply_derivatives_transpose(
              c:          NDArray,                # residual jet (scalar), shape=(order+1)+W+K+C
              ww:         typ.Sequence[NDArray],  # probe vectors X,        len=d, elm_shape=W+(Ni,)
              pp:         typ.Sequence[NDArray],  # perturbation vectors P, len=d, elm_shape=W+(Ni,)
              frame:       typ.Tuple[
                  typ.Sequence[NDArray], typ.Sequence[NDArray],
                  typ.Sequence[NDArray], typ.Sequence[NDArray],
              ],                                  # = T3Frame.data = (U, O, P, Q)
              order:      int,                    # highest derivative order
              sum_over_probes: bool = False,      # True: sum the sample stack W (the J^T r back-projection)
      ) -> typ.Tuple[
          typ.Tuple[NDArray, ...],  # dU_tildes (Tucker variation gradient)
          typ.Tuple[NDArray, ...],  # dG_tildes (TT variation gradient)
      ]:                                          # = T3Variations.data


   Transpose of :py:func:`tv_apply_derivatives`: back-project residual jets ``c`` into a
   variation gradient ``(dU_tildes, dG_tildes)``. The adjoint-state apply transpose -- the scalar
   residual jet ``c`` seeds one propagation sweep (no per-mode residual, no nu/eta), so it is about
   half the probe transpose. Full ``W + K + C`` stacking: ``c`` carries the tangent stack ``K``, which
   rides to the variation gradient; ``sum_over_probes`` sums (``True``, the Gauss-Newton ``J^T r``) or
   keeps (``False``, ``W`` rides into the variation stack) the sample stack ``W``; ``K``/``C`` always
   kept. Verified vs the dense adjoint identity ``<c, J v> = <J^T c, v>`` and ``jax.linear_transpose``.

   .. seealso:: :py:obj:`tv_apply_derivatives`, :py:obj:`tv_entries_derivatives_transpose`, :py:obj:`tv_probe_derivatives_transpose`