Grassmann Projections
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
A collection of methods to project data on the Grassmann manifold.

The abstract :class:`.GrassmannProjection` class is the parent class that allows the user to define methods for
projection of data onto the Grassmann manifold. These projection methods must be created within child classes built from
this abstract class.

.. autoclass:: UQpy.dimension_reduction.grassmann_manifold.projections.baseclass.GrassmannProjection
    :members:

The :class:`.GrassmannProjection` class is imported using the following command:

>>> from UQpy.dimension_reduction.grassmann_manifold.projections.baseclass.GrassmannProjection import GrassmannProjection

The user may create a new Grassmann projection method by subclassing :class:`.GrassmannProjection`. Any such class
must create points with type :class:`.GrassmannPoint`.

SVD Projection
~~~~~~~~~~~~~~~~~~~~~~

The :class:`.SVDProjection` class is used to project each data point of a given dataset onto a Grassmann manifold using
the Singular Value Decomposition (SVD). The SVD factorizes a matrix :math:`\mathbf{X}\in \mathbb{R}^{n \times m}` into
three matrices:

.. math:: \mathbf{X} =  \mathbf{U}  \mathbf{\Sigma}  \mathbf{V}^\intercal

where :math:`\mathbf{U}` and :math:`\mathbf{V}` are the matrices of left and right eigenvectors, respectively and
:math:`\mathbf{\Sigma}` is a diagonal matrix containing the eigenvalues. Since :math:`\mathbf{U}` and :math:`\mathbf{V}`
are orthonormal matrices we consider them to be representatives of the data point on the Grassmann manifold. The
:class:`.SvdProjection` class allows the user to define the Grassmann manifold :math:`\mathcal{G}(p, n)` on which the
data will reside by selecting the number of :math:`p-` planes, i.e., the rank of matrix :math:`\mathbf{U}` is equal to
the number of :math:`p-` planes.

The :class:`.SVDProjection` class is imported using the following command:

>>> from UQpy.dimension_reduction.SVDProjection import SVDProjection

A description of the class signature is shown below:

.. autoclass:: UQpy.dimension_reduction.SVDProjection

Attributes
~~~~~~~~~~~~~~~~~~
.. autoattribute:: UQpy.dimension_reduction.SVDProjection.u
.. autoattribute:: UQpy.dimension_reduction.SVDProjection.sigma
.. autoattribute:: UQpy.dimension_reduction.SVDProjection.v