.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery/grass.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_gallery_grass.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_gallery_grass.py:


.. _gallery:dict:grass:1:

Grassmannian Frames
=====================

.. contents::
    :depth: 2
    :local:

A Grassmannian (real) frame :math:`\Phi` is an :math:`m \times n` matrix (dictionary)
with unit norm columns such that the individual columns (atoms) are
as far away from each other as possible.
In other words, it is a matrix with minimum possible coherence.
If :math:`\bG = \Phi^T \Phi` then every off diagonal entry of
the Gram matrix :math:`\bG` has the value:

.. math::

    |g_{i j} | = \sqrt{\frac{n - m}{m (n - 1)}}

Grassmannian frames are hard to construct.
CR-Sparse library includes a method based on
alternate projections to construct a Grassmannian frame
starting from a random dictionary.

.. GENERATED FROM PYTHON SOURCE LINES 27-32

.. code-block:: default


    # Configure JAX to work with 64-bit floating point precision. 
    from jax.config import config
    config.update("jax_enable_x64", True)








.. GENERATED FROM PYTHON SOURCE LINES 33-34

Let's import necessary libraries 

.. GENERATED FROM PYTHON SOURCE LINES 34-43

.. code-block:: default


    import jax
    import numpy as np
    import jax.numpy as jnp
    import cr.nimble as crn
    import cr.sparse as crs
    import cr.sparse.dict as crdict
    from matplotlib import pyplot as plt








.. GENERATED FROM PYTHON SOURCE LINES 44-46

Frame size
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 46-50

.. code-block:: default

    m = 50
    n = 100









.. GENERATED FROM PYTHON SOURCE LINES 51-53

Minimum possible coherence
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 53-58

.. code-block:: default

    min_mu = crdict.minimum_coherence(m, n)
    print(f'Minimum coherence {min_mu:.5f}')







.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Minimum coherence 0.10050




.. GENERATED FROM PYTHON SOURCE LINES 59-61

Construction of Grassmannian frame
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 61-73

.. code-block:: default


    # Start with a Gaussian random dictionary
    init = crdict.gaussian_mtx(crn.KEYS[0], m, n)
    # Iteratively bring it close to a Grassmannian frame
    frame = crdict.build_grassmannian_frame(init, iterations=50)
    # The Gram matrix of the final frame
    gram = frame.T @ frame
    # Off diagonal elements of the gram matrix
    off = np.asarray(crn.off_diagonal_elements(gram))
    print(f'min: {np.min(off):.3f}, max: {np.max(off):.3f}, mean: {np.mean(np.abs(off)):.5f}')
    # Absolute values of the Gram matrix
    plt.imshow(np.abs(gram))



.. image-sg:: /gallery/images/sphx_glr_grass_001.png
   :alt: grass
   :srcset: /gallery/images/sphx_glr_grass_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    min: -0.160, max: 0.166, mean: 0.11393

    <matplotlib.image.AxesImage object at 0x7f27b7701310>




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 1.359 seconds)


.. _sphx_glr_download_gallery_grass.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example




    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: grass.py <grass.py>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: grass.ipynb <grass.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_