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+
+
+.. _sphx_glr_auto_examples_plot_OT_1D_smooth.py:
+
+
+===========================
+1D smooth optimal transport
+===========================
+
+This example illustrates the computation of EMD, Sinkhorn and smooth OT plans
+and their visualization.
+
+
+
+
+.. code-block:: python
+
+
+    # Author: Remi Flamary <remi.flamary@unice.fr>
+    #
+    # License: MIT License
+
+    import numpy as np
+    import matplotlib.pylab as pl
+    import ot
+    import ot.plot
+    from ot.datasets import make_1D_gauss as gauss
+
+
+
+
+
+
+
+Generate data
+-------------
+
+
+
+.. code-block:: python
+
+
+
+    #%% parameters
+
+    n = 100  # nb bins
+
+    # bin positions
+    x = np.arange(n, dtype=np.float64)
+
+    # Gaussian distributions
+    a = gauss(n, m=20, s=5)  # m= mean, s= std
+    b = gauss(n, m=60, s=10)
+
+    # loss matrix
+    M = ot.dist(x.reshape((n, 1)), x.reshape((n, 1)))
+    M /= M.max()
+
+
+
+
+
+
+
+
+Plot distributions and loss matrix
+----------------------------------
+
+
+
+.. code-block:: python
+
+
+    #%% plot the distributions
+
+    pl.figure(1, figsize=(6.4, 3))
+    pl.plot(x, a, 'b', label='Source distribution')
+    pl.plot(x, b, 'r', label='Target distribution')
+    pl.legend()
+
+    #%% plot distributions and loss matrix
+
+    pl.figure(2, figsize=(5, 5))
+    ot.plot.plot1D_mat(a, b, M, 'Cost matrix M')
+
+
+
+
+.. rst-class:: sphx-glr-horizontal
+
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_1D_smooth_001.png
+            :scale: 47
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_1D_smooth_002.png
+            :scale: 47
+
+
+
+
+Solve EMD
+---------
+
+
+
+.. code-block:: python
+
+
+
+    #%% EMD
+
+    G0 = ot.emd(a, b, M)
+
+    pl.figure(3, figsize=(5, 5))
+    ot.plot.plot1D_mat(a, b, G0, 'OT matrix G0')
+
+
+
+
+.. image:: /auto_examples/images/sphx_glr_plot_OT_1D_smooth_005.png
+    :align: center
+
+
+
+
+Solve Sinkhorn
+--------------
+
+
+
+.. code-block:: python
+
+
+
+    #%% Sinkhorn
+
+    lambd = 2e-3
+    Gs = ot.sinkhorn(a, b, M, lambd, verbose=True)
+
+    pl.figure(4, figsize=(5, 5))
+    ot.plot.plot1D_mat(a, b, Gs, 'OT matrix Sinkhorn')
+
+    pl.show()
+
+
+
+
+.. image:: /auto_examples/images/sphx_glr_plot_OT_1D_smooth_007.png
+    :align: center
+
+
+.. rst-class:: sphx-glr-script-out
+
+ Out::
+
+    It.  |Err         
+    -------------------
+        0|7.958844e-02|
+       10|5.921715e-03|
+       20|1.238266e-04|
+       30|2.469780e-06|
+       40|4.919966e-08|
+       50|9.800197e-10|
+
+
+Solve Smooth OT
+--------------
+
+
+
+.. code-block:: python
+
+
+
+    #%% Smooth OT with KL regularization
+
+    lambd = 2e-3
+    Gsm = ot.smooth.smooth_ot_dual(a, b, M, lambd, reg_type='kl')
+
+    pl.figure(5, figsize=(5, 5))
+    ot.plot.plot1D_mat(a, b, Gsm, 'OT matrix Smooth OT KL reg.')
+
+    pl.show()
+
+
+    #%% Smooth OT with KL regularization
+
+    lambd = 1e-1
+    Gsm = ot.smooth.smooth_ot_dual(a, b, M, lambd, reg_type='l2')
+
+    pl.figure(6, figsize=(5, 5))
+    ot.plot.plot1D_mat(a, b, Gsm, 'OT matrix Smooth OT l2 reg.')
+
+    pl.show()
+
+
+
+.. rst-class:: sphx-glr-horizontal
+
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_1D_smooth_009.png
+            :scale: 47
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_OT_1D_smooth_010.png
+            :scale: 47
+
+
+
+
+**Total running time of the script:** ( 0 minutes  1.053 seconds)
+
+
+
+.. only :: html
+
+ .. container:: sphx-glr-footer
+
+
+  .. container:: sphx-glr-download
+
+     :download:`Download Python source code: plot_OT_1D_smooth.py <plot_OT_1D_smooth.py>`
+
+
+
+  .. container:: sphx-glr-download
+
+     :download:`Download Jupyter notebook: plot_OT_1D_smooth.ipynb <plot_OT_1D_smooth.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_