UniformCorewiseGeometry
=======================

.. toctree::
   :hidden:

   /autoapi/t3toolbox/uniform_manifold/UniformCorewiseGeometry.frame
   /autoapi/t3toolbox/uniform_manifold/UniformCorewiseGeometry.randn
   /autoapi/t3toolbox/uniform_manifold/UniformCorewiseGeometry.randn_like
   /autoapi/t3toolbox/uniform_manifold/UniformCorewiseGeometry.project
   /autoapi/t3toolbox/uniform_manifold/UniformCorewiseGeometry.inner
   /autoapi/t3toolbox/uniform_manifold/UniformCorewiseGeometry.norm
   /autoapi/t3toolbox/uniform_manifold/UniformCorewiseGeometry.retract

.. py:class:: t3toolbox.uniform_manifold.UniformCorewiseGeometry

   The Euclidean geometry of the uniform core parameter space (uniform mirror of
   :py:class:`~t3toolbox.manifold.CorewiseGeometry`).

   Optimization happens *on* the raw supercores: tangents are perturbations of the cores ``(U, G, G, G)``
   (the non-orthonormal frame whose down/left/right cores are all the TT supercore ``G`` -- the Section 6.3
   ``(P, Q, O) -> G`` substitution), the metric is the plain Euclidean (corewise) inner product, the
   "projection" is the identity (no gauge), and the retraction is vector addition in the supercores. Use the
   module singleton :py:data:`UNIFORM_COREWISE`. The manifold counterpart is
   :py:class:`UniformManifoldGeometry`.

   .. 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_COREWISE.frame(x)            # the (U, G, G, G) frame
   >>> v = ut3m.UNIFORM_COREWISE.randn(frame)           # raw randn cores (no gauge)
   >>> print(bool(v.is_gauged().all()))
   False
   >>> y = ut3m.UNIFORM_COREWISE.retract(ut3m.UT3Tangent.zeros(frame))   # additive: cores += step (zero -> the point)
   >>> print(bool(np.allclose(y.to_dense(), x.to_dense())))
   True

Methods
-------

.. autoapisummary::

   t3toolbox.uniform_manifold.UniformCorewiseGeometry.frame
   t3toolbox.uniform_manifold.UniformCorewiseGeometry.randn
   t3toolbox.uniform_manifold.UniformCorewiseGeometry.randn_like
   t3toolbox.uniform_manifold.UniformCorewiseGeometry.project
   t3toolbox.uniform_manifold.UniformCorewiseGeometry.inner
   t3toolbox.uniform_manifold.UniformCorewiseGeometry.norm
   t3toolbox.uniform_manifold.UniformCorewiseGeometry.retract