.. only:: html
.. note::
:class: sphx-glr-download-link-note
Click :ref:`here <sphx_glr_download_auto_examples_plot_fgw.py>` to download the full example code
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_auto_examples_plot_fgw.py:
This example illustrates the computation of FGW for 1D measures[18].
| [18] |
"Optimal Transport for structured data with application on graphs" International Conference on Machine Learning (ICML). 2019. |
# Author: Titouan Vayer <titouan.vayer@irisa.fr>
#
# License: MIT License
import matplotlib.pyplot as pl
import numpy as np
import ot
from ot.gromov import gromov_wasserstein, fused_gromov_wasserstein
We create two 1D random measures
n = 20 # number of points in the first distribution
n2 = 30 # number of points in the second distribution
sig = 1 # std of first distribution
sig2 = 0.1 # std of second distribution
np.random.seed(0)
phi = np.arange(n)[:, None]
xs = phi + sig * np.random.randn(n, 1)
ys = np.vstack((np.ones((n // 2, 1)), 0 * np.ones((n // 2, 1)))) + sig2 * np.random.randn(n, 1)
phi2 = np.arange(n2)[:, None]
xt = phi2 + sig * np.random.randn(n2, 1)
yt = np.vstack((np.ones((n2 // 2, 1)), 0 * np.ones((n2 // 2, 1)))) + sig2 * np.random.randn(n2, 1)
yt = yt[::-1, :]
p = ot.unif(n)
q = ot.unif(n2)
pl.close(10)
pl.figure(10, (7, 7))
pl.subplot(2, 1, 1)
pl.scatter(ys, xs, c=phi, s=70)
pl.ylabel('Feature value a', fontsize=20)
pl.title('$\mu=\sum_i \delta_{x_i,a_i}$', fontsize=25, y=1)
pl.xticks(())
pl.yticks(())
pl.subplot(2, 1, 2)
pl.scatter(yt, xt, c=phi2, s=70)
pl.xlabel('coordinates x/y', fontsize=25)
pl.ylabel('Feature value b', fontsize=20)
pl.title('$\\nu=\sum_j \delta_{y_j,b_j}$', fontsize=25, y=1)
pl.yticks(())
pl.tight_layout()
pl.show()
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
/home/circleci/project/examples/plot_fgw.py:73: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
pl.show()
C1 = ot.dist(xs)
C2 = ot.dist(xt)
M = ot.dist(ys, yt)
w1 = ot.unif(C1.shape[0])
w2 = ot.unif(C2.shape[0])
Got = ot.emd([], [], M)
cmap = 'Reds'
pl.close(10)
pl.figure(10, (5, 5))
fs = 15
l_x = [0, 5, 10, 15]
l_y = [0, 5, 10, 15, 20, 25]
gs = pl.GridSpec(5, 5)
ax1 = pl.subplot(gs[3:, :2])
pl.imshow(C1, cmap=cmap, interpolation='nearest')
pl.title("$C_1$", fontsize=fs)
pl.xlabel("$k$", fontsize=fs)
pl.ylabel("$i$", fontsize=fs)
pl.xticks(l_x)
pl.yticks(l_x)
ax2 = pl.subplot(gs[:3, 2:])
pl.imshow(C2, cmap=cmap, interpolation='nearest')
pl.title("$C_2$", fontsize=fs)
pl.ylabel("$l$", fontsize=fs)
#pl.ylabel("$l$",fontsize=fs)
pl.xticks(())
pl.yticks(l_y)
ax2.set_aspect('auto')
ax3 = pl.subplot(gs[3:, 2:], sharex=ax2, sharey=ax1)
pl.imshow(M, cmap=cmap, interpolation='nearest')
pl.yticks(l_x)
pl.xticks(l_y)
pl.ylabel("$i$", fontsize=fs)
pl.title("$M_{AB}$", fontsize=fs)
pl.xlabel("$j$", fontsize=fs)
pl.tight_layout()
ax3.set_aspect('auto')
pl.show()
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
/home/circleci/project/examples/plot_fgw.py:128: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
pl.show()
alpha = 1e-3
ot.tic()
Gwg, logw = fused_gromov_wasserstein(M, C1, C2, p, q, loss_fun='square_loss', alpha=alpha, verbose=True, log=True)
ot.toc()
#%reload_ext WGW
Gg, log = gromov_wasserstein(C1, C2, p, q, loss_fun='square_loss', verbose=True, log=True)
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
It. |Loss |Relative loss|Absolute loss
------------------------------------------------
0|4.734412e+01|0.000000e+00|0.000000e+00
1|2.508254e+01|8.875326e-01|2.226158e+01
2|2.189327e+01|1.456740e-01|3.189279e+00
3|2.189327e+01|0.000000e+00|0.000000e+00
Elapsed time : 0.0013608932495117188 s
It. |Loss |Relative loss|Absolute loss
------------------------------------------------
0|4.683978e+04|0.000000e+00|0.000000e+00
1|3.860061e+04|2.134468e-01|8.239175e+03
2|2.182948e+04|7.682787e-01|1.677113e+04
3|2.182948e+04|0.000000e+00|0.000000e+00
cmap = 'Blues'
fs = 15
pl.figure(2, (13, 5))
pl.clf()
pl.subplot(1, 3, 1)
pl.imshow(Got, cmap=cmap, interpolation='nearest')
#pl.xlabel("$y$",fontsize=fs)
pl.ylabel("$i$", fontsize=fs)
pl.xticks(())
pl.title('Wasserstein ($M$ only)')
pl.subplot(1, 3, 2)
pl.imshow(Gg, cmap=cmap, interpolation='nearest')
pl.title('Gromov ($C_1,C_2$ only)')
pl.xticks(())
pl.subplot(1, 3, 3)
pl.imshow(Gwg, cmap=cmap, interpolation='nearest')
pl.title('FGW ($M+C_1,C_2$)')
pl.xlabel("$j$", fontsize=fs)
pl.ylabel("$i$", fontsize=fs)
pl.tight_layout()
pl.show()
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
/home/circleci/project/examples/plot_fgw.py:173: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
pl.show()
.. rst-class:: sphx-glr-timing
**Total running time of the script:** ( 0 minutes 0.514 seconds)
.. only :: html
.. container:: sphx-glr-footer
:class: sphx-glr-footer-example
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: plot_fgw.py <plot_fgw.py>`
.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: plot_fgw.ipynb <plot_fgw.ipynb>`
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_