UniformManifoldGeometry
=======================

.. toctree::
   :hidden:

   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.frame
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.randn
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.random_orthogonal
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.randn_like
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.project
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.project_oblique
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.inner
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.norm
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.retract
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.project_ambient
   /autoapi/t3toolbox/uniform_manifold/UniformManifoldGeometry.transport

.. py:class:: t3toolbox.uniform_manifold.UniformManifoldGeometry

   The Riemannian geometry of the uniform fixed-rank Tucker tensor train manifold (uniform mirror of
   :py:class:`~t3toolbox.manifold.ManifoldGeometry`).

   A stateless bundle of the chart-level choices that distinguish the manifold from the over-parametrized
   corewise geometry -- the orthonormal frame (:py:meth:`frame`), the gauge projection ``Pi``
   (:py:meth:`project`), and the manifold retraction (:py:meth:`retract`) -- plus the ambient projection and
   transport. Tangents live in ``T_xM`` with an orthonormal, gauged frame; the metric is Hilbert-Schmidt.
   All methods are **per-element-mask-aware** (varying ranks across the frame stack ``C`` are supported) and
   carry the per-element safe-mode preconditions (orthogonal frame, gauged variations; ``.all()`` over the
   stack). Use the module singleton :py:data:`UNIFORM_MANIFOLD`. The corewise counterpart is
   :py:class:`UniformCorewiseGeometry`.

   .. rubric:: Examples

   >>> import numpy as np
   >>> import t3toolbox.tucker_tensor_train as t3
   >>> import t3toolbox.uniform_tucker_tensor_train as ut3
   >>> import t3toolbox.uniform_manifold as ut3m
   >>> np.random.seed(0)
   >>> x = ut3.UniformTuckerTensorTrain.from_t3(t3.TuckerTensorTrain.randn((10, 11, 12), (3, 4, 3), (1, 2, 2, 1)))
   >>> frame = ut3m.UNIFORM_MANIFOLD.frame(x)            # orthonormal frame at x
   >>> print(bool(frame.is_orthogonal().all()))
   True
   >>> v = ut3m.UNIFORM_MANIFOLD.randn(frame)           # a standard Gaussian on T_xM (gauged)
   >>> print(bool(v.is_gauged().all()))
   True
   >>> y = ut3m.UNIFORM_MANIFOLD.retract(ut3m.UT3Tangent.zeros(frame))   # retract the zero tangent == the base point
   >>> print(bool(np.allclose(y.to_dense(), x.to_dense())))
   True

Methods
-------

.. autoapisummary::

   t3toolbox.uniform_manifold.UniformManifoldGeometry.frame
   t3toolbox.uniform_manifold.UniformManifoldGeometry.randn
   t3toolbox.uniform_manifold.UniformManifoldGeometry.random_orthogonal
   t3toolbox.uniform_manifold.UniformManifoldGeometry.randn_like
   t3toolbox.uniform_manifold.UniformManifoldGeometry.project
   t3toolbox.uniform_manifold.UniformManifoldGeometry.project_oblique
   t3toolbox.uniform_manifold.UniformManifoldGeometry.inner
   t3toolbox.uniform_manifold.UniformManifoldGeometry.norm
   t3toolbox.uniform_manifold.UniformManifoldGeometry.retract
   t3toolbox.uniform_manifold.UniformManifoldGeometry.project_ambient
   t3toolbox.uniform_manifold.UniformManifoldGeometry.transport