ManifoldGeometryOps
===================

.. toctree::
   :hidden:

   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.with_sharing
   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.frame
   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.stack_shape
   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.base_point
   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.precompute
   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.project
   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.retract
   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.inner
   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.point_norm_sq
   /autoapi/t3toolbox/backend/geometry/ManifoldGeometryOps.point_tangent

.. py:class:: t3toolbox.backend.geometry.ManifoldGeometryOps

   Bases: :py:obj:`t3toolbox.backend.common.ValueHashedFields`


   The fixed-rank T3 manifold geometry on raw ``(tucker_cores, tt_cores)`` data -- the check-free twin
   of :py:data:`t3toolbox.manifold.MANIFOLD`.

   ``frame`` is the orthonormal frame (Algorithm 11), ``project`` the gauge projection ``Pi``, ``retract``
   the implicit truncated T3-SVD retraction, and ``inner`` the ragged coordinate dot (equal to
   Hilbert-Schmidt on an orthonormal, gauged frame). With a non-empty ``groups`` this is the SF-T3
   geometry: ``precompute`` derives the per-frame companion
   (:py:func:`~t3toolbox.backend.sharing.fv_shared_frame_data`, built once per local model),
   ``project`` composes the gauge projection with the tied post-pass, and ``retract`` builds the tied
   doubled-rank embedding and truncates with the grouped ``t3svd``. ``inner`` / ``point_norm_sq`` /
   ``point_tangent`` are unaffected by tying (the tied subspace is linear, so the restriction of the
   metric is itself, and the base-point tangent has zero Tucker variations, hence is trivially tied).


   .. py:attribute:: groups
      :type:  t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.typ.Tuple[int, ...], ...]
      :value: ()


Methods
-------

.. autoapisummary::

   t3toolbox.backend.geometry.ManifoldGeometryOps.with_sharing
   t3toolbox.backend.geometry.ManifoldGeometryOps.frame
   t3toolbox.backend.geometry.ManifoldGeometryOps.stack_shape
   t3toolbox.backend.geometry.ManifoldGeometryOps.base_point
   t3toolbox.backend.geometry.ManifoldGeometryOps.precompute
   t3toolbox.backend.geometry.ManifoldGeometryOps.project
   t3toolbox.backend.geometry.ManifoldGeometryOps.retract
   t3toolbox.backend.geometry.ManifoldGeometryOps.inner
   t3toolbox.backend.geometry.ManifoldGeometryOps.point_norm_sq
   t3toolbox.backend.geometry.ManifoldGeometryOps.point_tangent