update Readme + minor rendering in examples

Author: Hicham Janati

README.md

@@ -27,6 +27,7 @@ It provides the following solvers:
 * Gromov-Wasserstein distances and barycenters ([13] and regularized [12])
 * Stochastic Optimization for Large-scale Optimal Transport (semi-dual problem [18] and dual problem [19])
 * Non regularized free support Wasserstein barycenters [20].
+* Unbalanced OT with KL relaxation distance and barycenter [10, 25].
 
 Some demonstrations (both in Python and Jupyter Notebook format) are available in the examples folder.
 
@@ -165,6 +166,7 @@ The contributors to this library are:
 * [Kilian Fatras](https://kilianfatras.github.io/)
 * [Alain Rakotomamonjy](https://sites.google.com/site/alainrakotomamonjy/home)
 * [Vayer Titouan](https://tvayer.github.io/)
+* [Hicham Janati](https://hichamjanati.github.io/) (Unbalanced OT)
 
 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):
 
@@ -236,3 +238,5 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 [23] Aude, G., Peyré, G., Cuturi, M., [Learning Generative Models with Sinkhorn Divergences](https://arxiv.org/abs/1706.00292), Proceedings of the Twenty-First International Conference on Artficial Intelligence and Statistics, (AISTATS) 21, 2018
 
 [24] Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N. (2019). [Optimal Transport for structured data with application on graphs](http://proceedings.mlr.press/v97/titouan19a.html) Proceedings of the 36th International Conference on Machine Learning (ICML).
+
+[25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. (2019). [Learning with a Wasserstein Loss](http://cbcl.mit.edu/wasserstein/)  Advances in Neural Information Processing Systems (NIPS).

examples/plot_UOT_1D.py

@@ -1,8 +1,8 @@
 # -*- coding: utf-8 -*-
 """
-====================
+===============================
 1D Unbalanced optimal transport
-====================
+===============================
 
 This example illustrates the computation of Unbalanced Optimal transport
 using a Kullback-Leibler relaxation.
@@ -53,7 +53,7 @@
 pl.plot(x, b, 'r', label='Target distribution')
 pl.legend()
 
-#%% plot distributions and loss matrix
+# plot distributions and loss matrix
 
 pl.figure(2, figsize=(5, 5))
 ot.plot.plot1D_mat(a, b, M, 'Cost matrix M')
@@ -64,7 +64,7 @@
 # --------------
 
 
-#%% Sinkhorn
+# Sinkhorn
 
 epsilon = 0.1  # entropy parameter
 alpha = 1.  # Unbalanced KL relaxation parameter

examples/plot_UOT_barycenter_1D.py

@@ -27,7 +27,7 @@
 # Generate data
 # -------------
 
-#%% parameters
+# parameters
 
 n = 100  # nb bins
 
@@ -53,7 +53,7 @@
 # Plot data
 # ---------
 
-#%% plot the distributions
+# plot the distributions
 
 pl.figure(1, figsize=(6.4, 3))
 for i in range(n_distributions):
@@ -65,7 +65,7 @@
 # Barycenter computation
 # ----------------------
 
-#%% non weighted barycenter computation
+# non weighted barycenter computation
 
 weight = 0.5  # 0<=weight<=1
 weights = np.array([1 - weight, weight])
@@ -97,7 +97,7 @@
 # Barycentric interpolation
 # -------------------------
 
-#%% barycenter interpolation
+# barycenter interpolation
 
 n_weight = 11
 weight_list = np.linspace(0, 1, n_weight)
@@ -114,7 +114,7 @@
     B_wass[:, i] = ot.unbalanced.barycenter_unbalanced(A, M, reg, alpha, weights)
 
 
-#%% plot interpolation
+# plot interpolation
 
 pl.figure(3)
 

ot/unbalanced.py

@@ -87,7 +87,7 @@ def sinkhorn_unbalanced(a, b, M, reg, alpha, method='sinkhorn', numItermax=1000,
 
     .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.
 
-    .. [23] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss,  Advances in Neural Information Processing Systems (NIPS) 2015
+    .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss,  Advances in Neural Information Processing Systems (NIPS) 2015
 
 
     See Also
@@ -196,7 +196,7 @@ def sinkhorn_unbalanced2(a, b, M, reg, alpha, method='sinkhorn',
 
     .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.
 
-    .. [23] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss,  Advances in Neural Information Processing Systems (NIPS) 2015
+    .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss,  Advances in Neural Information Processing Systems (NIPS) 2015
 
     See Also
     --------
@@ -299,7 +299,7 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000,
 
     .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.
 
-    .. [23] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss,  Advances in Neural Information Processing Systems (NIPS) 2015
+    .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss,  Advances in Neural Information Processing Systems (NIPS) 2015
 
     See Also
     --------