UniformManifoldGeometry.project_ambient
=======================================

.. py:method:: t3toolbox.uniform_manifold.UniformManifoldGeometry.project_ambient(frame, grad)

   .. code-block:: python

      def project_ambient(
              self,
              frame:  ubv.UT3Frame,                     # orthogonal base point of the tangent space
              grad:   ut3.UniformTuckerTensorTrain,     # ambient gradient as a uniform Tucker tensor train
      ) -> UT3Tangent:  # the Riemannian gradient (gauged projection of grad) at frame


   Project an ambient gradient onto ``T_xM`` -- the Riemannian gradient.

   ``grad`` is the Euclidean/ambient gradient as a :py:class:`UniformTuckerTensorTrain`. Returns the
   gauged tangent ``P_T(grad)`` (the residual ``grad - P_T(grad)`` is orthogonal to the tangent space).
   Requires an **orthogonal** ``frame`` (minimal rank not required); enforced in safe mode (skipped under
   ``safety.unsafe()`` / a jax trace).

   Unlike the ragged :py:meth:`~t3toolbox.manifold.ManifoldGeometry.project_ambient`, the uniform layer
   has **no native dense-array path** (the uniform layer is a performance layer; working dense here
   defeats its purpose). For a dense gradient, go via the ragged geometry and the cross-layer converters
   (:py:meth:`UT3Frame.to_t3frame` -> ``MANIFOLD.project_ambient`` -> :py:meth:`UT3Tangent.from_t3tangent`).