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+
+
+.. _sphx_glr_auto_examples_plot_OT_2D_samples.py:
+
+
+====================================================
+2D Optimal transport between empirical distributions
+====================================================
+
+Illustration of 2D optimal transport between discributions that are weighted
+sum of diracs. The OT matrix is plotted with the samples.
+
+
+
+
+.. code-block:: python
+
+
+    # Author: Remi Flamary <remi.flamary@unice.fr>
+    #         Kilian Fatras <kilian.fatras@irisa.fr>
+    #
+    # License: MIT License
+
+    import numpy as np
+    import matplotlib.pylab as pl
+    import ot
+    import ot.plot
+
+
+
+
+
+
+
+Generate data
+-------------
+
+
+
+.. code-block:: python
+
+
+    #%% parameters and data generation
+
+    n = 50  # nb samples
+
+    mu_s = np.array([0, 0])
+    cov_s = np.array([[1, 0], [0, 1]])
+
+    mu_t = np.array([4, 4])
+    cov_t = np.array([[1, -.8], [-.8, 1]])
+
+    xs = ot.datasets.make_2D_samples_gauss(n, mu_s, cov_s)
+    xt = ot.datasets.make_2D_samples_gauss(n, mu_t, cov_t)
+
+    a, b = np.ones((n,)) / n, np.ones((n,)) / n  # uniform distribution on samples
+
+    # loss matrix
+    M = ot.dist(xs, xt)
+    M /= M.max()
+
+
+
+
+
+
+
+Plot data
+---------
+
+
+
+.. code-block:: python
+
+
+    #%% plot samples
+
+    pl.figure(1)
+    pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+    pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+    pl.legend(loc=0)
+    pl.title('Source and target distributions')
+
+    pl.figure(2)
+    pl.imshow(M, interpolation='nearest')
+    pl.title('Cost matrix M')
+
+
+
+
+.. rst-class:: sphx-glr-horizontal
+
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_001.png
+            :scale: 47
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_002.png
+            :scale: 47
+
+
+
+
+Compute EMD
+-----------
+
+
+
+.. code-block:: python
+
+
+    #%% EMD
+
+    G0 = ot.emd(a, b, M)
+
+    pl.figure(3)
+    pl.imshow(G0, interpolation='nearest')
+    pl.title('OT matrix G0')
+
+    pl.figure(4)
+    ot.plot.plot2D_samples_mat(xs, xt, G0, c=[.5, .5, 1])
+    pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+    pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+    pl.legend(loc=0)
+    pl.title('OT matrix with samples')
+
+
+
+
+
+.. rst-class:: sphx-glr-horizontal
+
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_005.png
+            :scale: 47
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_006.png
+            :scale: 47
+
+
+
+
+Compute Sinkhorn
+----------------
+
+
+
+.. code-block:: python
+
+
+    #%% sinkhorn
+
+    # reg term
+    lambd = 1e-3
+
+    Gs = ot.sinkhorn(a, b, M, lambd)
+
+    pl.figure(5)
+    pl.imshow(Gs, interpolation='nearest')
+    pl.title('OT matrix sinkhorn')
+
+    pl.figure(6)
+    ot.plot.plot2D_samples_mat(xs, xt, Gs, color=[.5, .5, 1])
+    pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+    pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+    pl.legend(loc=0)
+    pl.title('OT matrix Sinkhorn with samples')
+
+    pl.show()
+
+
+
+
+
+.. rst-class:: sphx-glr-horizontal
+
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_009.png
+            :scale: 47
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_010.png
+            :scale: 47
+
+
+
+
+Emprirical Sinkhorn
+----------------
+
+
+
+.. code-block:: python
+
+
+    #%% sinkhorn
+
+    # reg term
+    lambd = 1e-3
+
+    Ges = ot.bregman.empirical_sinkhorn(xs, xt, lambd)
+
+    pl.figure(7)
+    pl.imshow(Ges, interpolation='nearest')
+    pl.title('OT matrix empirical sinkhorn')
+
+    pl.figure(8)
+    ot.plot.plot2D_samples_mat(xs, xt, Ges, color=[.5, .5, 1])
+    pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+    pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+    pl.legend(loc=0)
+    pl.title('OT matrix Sinkhorn from samples')
+
+    pl.show()
+
+
+
+.. rst-class:: sphx-glr-horizontal
+
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_013.png
+            :scale: 47
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_014.png
+            :scale: 47
+
+
+.. rst-class:: sphx-glr-script-out
+
+ Out::
+
+    Warning: numerical errors at iteration 0
+
+
+**Total running time of the script:** ( 0 minutes  2.616 seconds)
+
+
+
+.. only :: html
+
+ .. container:: sphx-glr-footer
+
+
+  .. container:: sphx-glr-download
+
+     :download:`Download Python source code: plot_OT_2D_samples.py <plot_OT_2D_samples.py>`
+
+
+
+  .. container:: sphx-glr-download
+
+     :download:`Download Jupyter notebook: plot_OT_2D_samples.ipynb <plot_OT_2D_samples.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_