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plot_OT_L1_vs_L2.rst.txt

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.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_plot_OT_L1_vs_L2.py>`
        to download the full example code

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

Optimal Transport with different ground metrics

2D OT on empirical distributio with different ground metric.

Stole the figure idea from Fig. 1 and 2 in https://arxiv.org/pdf/1706.07650.pdf

# Author: Remi Flamary <[email protected]>
#
# License: MIT License

# sphinx_gallery_thumbnail_number = 3

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

Dataset 1 : uniform sampling

n = 20  # nb samples
xs = np.zeros((n, 2))
xs[:, 0] = np.arange(n) + 1
xs[:, 1] = (np.arange(n) + 1) * -0.001  # to make it strictly convex...

xt = np.zeros((n, 2))
xt[:, 1] = np.arange(n) + 1

a, b = ot.unif(n), ot.unif(n)  # uniform distribution on samples

# loss matrix
M1 = ot.dist(xs, xt, metric='euclidean')
M1 /= M1.max()

# loss matrix
M2 = ot.dist(xs, xt, metric='sqeuclidean')
M2 /= M2.max()

# loss matrix
Mp = ot.dist(xs, xt, metric='cityblock')
Mp /= Mp.max()

# Data
pl.figure(1, figsize=(7, 3))
pl.clf()
pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
pl.axis('equal')
pl.title('Source and target distributions')


# Cost matrices
pl.figure(2, figsize=(7, 3))

pl.subplot(1, 3, 1)
pl.imshow(M1, interpolation='nearest')
pl.title('Euclidean cost')

pl.subplot(1, 3, 2)
pl.imshow(M2, interpolation='nearest')
pl.title('Squared Euclidean cost')

pl.subplot(1, 3, 3)
pl.imshow(Mp, interpolation='nearest')
pl.title('L1 (cityblock cost')
pl.tight_layout()

.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_001.png
         :alt: Source and target distributions
         :srcset: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_002.png
         :alt: Euclidean cost, Squared Euclidean cost, L1 (cityblock cost
         :srcset: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_002.png
         :class: sphx-glr-multi-img

Dataset 1 : Plot OT Matrices

G1 = ot.emd(a, b, M1)
G2 = ot.emd(a, b, M2)
Gp = ot.emd(a, b, Mp)

# OT matrices
pl.figure(3, figsize=(7, 3))

pl.subplot(1, 3, 1)
ot.plot.plot2D_samples_mat(xs, xt, G1, 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.axis('equal')
# pl.legend(loc=0)
pl.title('OT Euclidean')

pl.subplot(1, 3, 2)
ot.plot.plot2D_samples_mat(xs, xt, G2, 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.axis('equal')
# pl.legend(loc=0)
pl.title('OT squared Euclidean')

pl.subplot(1, 3, 3)
ot.plot.plot2D_samples_mat(xs, xt, Gp, 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.axis('equal')
# pl.legend(loc=0)
pl.title('OT L1 (cityblock)')
pl.tight_layout()

pl.show()

.. image-sg:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_003.png
   :alt: OT Euclidean, OT squared Euclidean, OT L1 (cityblock)
   :srcset: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_003.png
   :class: sphx-glr-single-img

Dataset 2 : Partial circle

n = 20  # nb samples
xtot = np.zeros((n + 1, 2))
xtot[:, 0] = np.cos(
    (np.arange(n + 1) + 1.0) * 0.8 / (n + 2) * 2 * np.pi)
xtot[:, 1] = np.sin(
    (np.arange(n + 1) + 1.0) * 0.8 / (n + 2) * 2 * np.pi)

xs = xtot[:n, :]
xt = xtot[1:, :]

a, b = ot.unif(n), ot.unif(n)  # uniform distribution on samples

# loss matrix
M1 = ot.dist(xs, xt, metric='euclidean')
M1 /= M1.max()

# loss matrix
M2 = ot.dist(xs, xt, metric='sqeuclidean')
M2 /= M2.max()

# loss matrix
Mp = ot.dist(xs, xt, metric='cityblock')
Mp /= Mp.max()


# Data
pl.figure(4, figsize=(7, 3))
pl.clf()
pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
pl.axis('equal')
pl.title('Source and target distributions')


# Cost matrices
pl.figure(5, figsize=(7, 3))

pl.subplot(1, 3, 1)
pl.imshow(M1, interpolation='nearest')
pl.title('Euclidean cost')

pl.subplot(1, 3, 2)
pl.imshow(M2, interpolation='nearest')
pl.title('Squared Euclidean cost')

pl.subplot(1, 3, 3)
pl.imshow(Mp, interpolation='nearest')
pl.title('L1 (cityblock) cost')
pl.tight_layout()

.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_004.png
         :alt: Source and target distributions
         :srcset: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_004.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_005.png
         :alt: Euclidean cost, Squared Euclidean cost, L1 (cityblock) cost
         :srcset: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_005.png
         :class: sphx-glr-multi-img

Dataset 2 : Plot OT Matrices

G1 = ot.emd(a, b, M1)
G2 = ot.emd(a, b, M2)
Gp = ot.emd(a, b, Mp)

# OT matrices
pl.figure(6, figsize=(7, 3))

pl.subplot(1, 3, 1)
ot.plot.plot2D_samples_mat(xs, xt, G1, 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.axis('equal')
# pl.legend(loc=0)
pl.title('OT Euclidean')

pl.subplot(1, 3, 2)
ot.plot.plot2D_samples_mat(xs, xt, G2, 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.axis('equal')
# pl.legend(loc=0)
pl.title('OT squared Euclidean')

pl.subplot(1, 3, 3)
ot.plot.plot2D_samples_mat(xs, xt, Gp, 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.axis('equal')
# pl.legend(loc=0)
pl.title('OT L1 (cityblock)')
pl.tight_layout()

pl.show()

.. image-sg:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_006.png
   :alt: OT Euclidean, OT squared Euclidean, OT L1 (cityblock)
   :srcset: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_006.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  1.319 seconds)


.. only:: html

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




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

      :download:`Download Python source code: plot_OT_L1_vs_L2.py <plot_OT_L1_vs_L2.py>`

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

      :download:`Download Jupyter notebook: plot_OT_L1_vs_L2.ipynb <plot_OT_L1_vs_L2.ipynb>`


.. only:: html

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

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