[MRG] Implementation of two news algorithms: SaGroW and PoGroW. (#275)

* Add two new algorithms to solve Gromov Wasserstein: Sampled Gromov Wasserstein and Pointwise Gromov Wasserstein.

* Correct some lines in SaGroW and PoGroW to follow pep8 guide.

* Change nb_samples name. Use rdm state. Change symmetric check.

* Change names of len(p) and len(q) in SaGroW and PoGroW.

* Re-add some deleted lines in the comments of gromov.py

Co-authored-by: Rémi Flamary <remi.flamary@gmail.com>

Author: Hv0nnus

README.md

@@ -28,6 +28,7 @@ POT provides the following generic OT solvers (links to examples):
 * [Gromov-Wasserstein distances](https://pythonot.github.io/auto_examples/gromov/plot_gromov.html) and [GW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_gromov_barycenter.html)  (exact [13] and regularized [12])
  * [Fused-Gromov-Wasserstein distances solver](https://pythonot.github.io/auto_examples/gromov/plot_fgw.html#sphx-glr-auto-examples-plot-fgw-py) and [FGW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_barycenter_fgw.html) [24]
 * [Stochastic solver](https://pythonot.github.io/auto_examples/plot_stochastic.html) for Large-scale Optimal Transport (semi-dual problem [18] and dual problem [19])
+* [Stochastic solver of Gromov Wasserstein](https://pythonot.github.io/auto_examples/gromov/plot_gromov.html) for large-scale problem with any loss functions [33]
 * Non regularized [free support Wasserstein barycenters](https://pythonot.github.io/auto_examples/barycenters/plot_free_support_barycenter.html) [20].
 * [Unbalanced OT](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_UOT_1D.html) with KL relaxation and [barycenter](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_UOT_barycenter_1D.html) [10, 25].
 * [Partial Wasserstein and Gromov-Wasserstein](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_partial_wass_and_gromov.html) (exact [29] and entropic [3]
@@ -198,6 +199,7 @@ The contributors to this library are
 * [Mokhtar Z. Alaya](http://mzalaya.github.io/) (Screenkhorn)
 * [Ievgen Redko](https://ievred.github.io/) (Laplacian DA, JCPOT)
 * [Adrien Corenflos](https://adriencorenflos.github.io/) (Sliced Wasserstein Distance)
+* [Tanguy Kerdoncuff](https://hv0nnus.github.io/) (Sampled Gromov Wasserstein)
 * [Minhui Huang](https://mhhuang95.github.io) (Projection Robust Wasserstein Distance)
 
 This toolbox benefit a lot from open source research and we would like to thank the following persons for providing some code (in various languages):
@@ -286,3 +288,5 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 [31] Bonneel, Nicolas, et al. [Sliced and radon wasserstein barycenters of measures](https://perso.liris.cnrs.fr/nicolas.bonneel/WassersteinSliced-JMIV.pdf), Journal of Mathematical Imaging and Vision 51.1 (2015): 22-45
 
 [32] Huang, M., Ma S., Lai, L. (2021). [A Riemannian Block Coordinate Descent Method for Computing the Projection Robust Wasserstein Distance](http://proceedings.mlr.press/v139/huang21e.html), Proceedings of the 38th International Conference on Machine Learning (ICML).
+
+[33] Kerdoncuff T., Emonet R., Marc S. [Sampled Gromov Wasserstein](https://hal.archives-ouvertes.fr/hal-03232509/document), Machine Learning Journal (MJL), 2021

examples/gromov/plot_gromov.py

@@ -104,3 +104,37 @@
 pl.title('Entropic Gromov Wasserstein')
 
 pl.show()
+
+#############################################################################
+#
+# Compute GW with a scalable stochastic method with any loss function
+# ----------------------------------------------------------------------
+
+
+def loss(x, y):
+    return np.abs(x - y)
+
+
+pgw, plog = ot.gromov.pointwise_gromov_wasserstein(C1, C2, p, q, loss, max_iter=100,
+                                                   log=True)
+
+sgw, slog = ot.gromov.sampled_gromov_wasserstein(C1, C2, p, q, loss, epsilon=0.1, max_iter=100,
+                                                 log=True)
+
+print('Pointwise Gromov-Wasserstein distance estimated: ' + str(plog['gw_dist_estimated']))
+print('Variance estimated: ' + str(plog['gw_dist_std']))
+print('Sampled Gromov-Wasserstein distance: ' + str(slog['gw_dist_estimated']))
+print('Variance estimated: ' + str(slog['gw_dist_std']))
+
+
+pl.figure(1, (10, 5))
+
+pl.subplot(1, 2, 1)
+pl.imshow(pgw.toarray(), cmap='jet')
+pl.title('Pointwise Gromov Wasserstein')
+
+pl.subplot(1, 2, 2)
+pl.imshow(sgw, cmap='jet')
+pl.title('Sampled Gromov Wasserstein')
+
+pl.show()