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

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_unbalanced-partial_plot_unbalanced_OT.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_unbalanced-partial_plot_unbalanced_OT.py:


==============================================================
2D examples of exact and entropic unbalanced optimal transport
==============================================================
This example is designed to show how to compute unbalanced and
partial OT in POT.

UOT aims at solving the following optimization problem:

    .. math::
        W = \min_{\gamma} <\gamma, \mathbf{M}>_F +
        \mathrm{reg}\cdot\Omega(\gamma) +
          \mathrm{reg_m} \cdot \mathrm{div}(\gamma \mathbf{1}, \mathbf{a}) +
        \mathrm{reg_m} \cdot \mathrm{div}(\gamma^T \mathbf{1}, \mathbf{b})

        s.t.
             \gamma \geq 0

where :math:`\mathrm{div}` is a divergence.
When using the entropic UOT, :math:`\mathrm{reg}>0` and :math:`\mathrm{div}`
should be the Kullback-Leibler divergence.
When solving exact UOT, :math:`\mathrm{reg}=0` and :math:`\mathrm{div}`
can be either the Kullback-Leibler or the quadratic divergence.
Using :math:`\ell_1` norm gives the so-called partial OT.

.. GENERATED FROM PYTHON SOURCE LINES 27-35

.. code-block:: Python


    # Author: Laetitia Chapel <laetitia.chapel@univ-ubs.fr>
    # License: MIT License

    import numpy as np
    import matplotlib.pylab as pl
    import ot








.. GENERATED FROM PYTHON SOURCE LINES 36-38

Generate data
-------------

.. GENERATED FROM PYTHON SOURCE LINES 40-67

.. code-block:: Python


    n = 40  # nb samples

    mu_s = np.array([-1, -1])
    cov_s = np.array([[1, 0], [0, 1]])

    mu_t = np.array([4, 4])
    cov_t = np.array([[1, -.8], [-.8, 1]])

    np.random.seed(0)
    xs = ot.datasets.make_2D_samples_gauss(n, mu_s, cov_s)
    xt = ot.datasets.make_2D_samples_gauss(n, mu_t, cov_t)

    n_noise = 10

    xs = np.concatenate((xs, ((np.random.rand(n_noise, 2) - 4))), axis=0)
    xt = np.concatenate((xt, ((np.random.rand(n_noise, 2) + 6))), axis=0)

    n = n + n_noise

    a, b = np.ones((n,)) / n, np.ones((n,)) / n  # uniform distribution on samples

    # loss matrix
    M = ot.dist(xs, xt)
    M /= M.max()









.. GENERATED FROM PYTHON SOURCE LINES 68-70

Compute entropic kl-regularized UOT, kl- and l2-regularized UOT
-----------

.. GENERATED FROM PYTHON SOURCE LINES 70-81

.. code-block:: Python


    reg = 0.005
    reg_m_kl = 0.05
    reg_m_l2 = 5
    mass = 0.7

    entropic_kl_uot = ot.unbalanced.sinkhorn_unbalanced(a, b, M, reg, reg_m_kl)
    kl_uot = ot.unbalanced.mm_unbalanced(a, b, M, reg_m_kl, div='kl')
    l2_uot = ot.unbalanced.mm_unbalanced(a, b, M, reg_m_l2, div='l2')
    partial_ot = ot.partial.partial_wasserstein(a, b, M, m=mass)








.. GENERATED FROM PYTHON SOURCE LINES 82-84

Plot the results
----------------

.. GENERATED FROM PYTHON SOURCE LINES 84-117

.. code-block:: Python


    pl.figure(2)
    transp = [partial_ot, l2_uot, kl_uot, entropic_kl_uot]
    title = ["partial OT \n m=" + str(mass), "$\ell_2$-UOT \n $\mathrm{reg_m}$=" +
             str(reg_m_l2), "kl-UOT \n $\mathrm{reg_m}$=" + str(reg_m_kl),
             "entropic kl-UOT \n $\mathrm{reg_m}$=" + str(reg_m_kl)]

    for p in range(4):
        pl.subplot(2, 4, p + 1)
        P = transp[p]
        if P.sum() > 0:
            P = P / P.max()
        for i in range(n):
            for j in range(n):
                if P[i, j] > 0:
                    pl.plot([xs[i, 0], xt[j, 0]], [xs[i, 1], xt[j, 1]], color='C2',
                            alpha=P[i, j] * 0.3)
        pl.scatter(xs[:, 0], xs[:, 1], c='C0', alpha=0.2)
        pl.scatter(xt[:, 0], xt[:, 1], c='C1', alpha=0.2)
        pl.scatter(xs[:, 0], xs[:, 1], c='C0', s=P.sum(1).ravel() * (1 + p) * 2)
        pl.scatter(xt[:, 0], xt[:, 1], c='C1', s=P.sum(0).ravel() * (1 + p) * 2)
        pl.title(title[p])
        pl.yticks(())
        pl.xticks(())
        if p < 1:
            pl.ylabel("mappings")
        pl.subplot(2, 4, p + 5)
        pl.imshow(P, cmap='jet')
        pl.yticks(())
        pl.xticks(())
        if p < 1:
            pl.ylabel("transport plans")
    pl.show()



.. image-sg:: /auto_examples/unbalanced-partial/images/sphx_glr_plot_unbalanced_OT_001.png
   :alt: partial OT   m=0.7, $\ell_2$-UOT   $\mathrm{reg_m}$=5, kl-UOT   $\mathrm{reg_m}$=0.05, entropic kl-UOT   $\mathrm{reg_m}$=0.05
   :srcset: /auto_examples/unbalanced-partial/images/sphx_glr_plot_unbalanced_OT_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 4.046 seconds)


.. _sphx_glr_download_auto_examples_unbalanced-partial_plot_unbalanced_OT.py:

.. only:: html

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

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

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

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

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


.. only:: html

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

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