Smooth and Sparse OT

README.md

@@ -15,7 +15,8 @@ It provides the following solvers:
 
 * OT Network Flow solver for the linear program/ Earth Movers Distance [1].
 * Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] and stabilized version [9][10] with optional GPU implementation (requires cudamat).
-* Non regularized Wasserstein barycenters [16] with LP solver.
+* Smooth optimal transport solvers (dual and semi-dual) for KL and squared L2 regularization [17].
+* Non regularized Wasserstein barycenters [16] with LP solver (only small scale).
 * Bregman projections for Wasserstein barycenter [3] and unmixing [4].
 * Optimal transport for domain adaptation with group lasso regularization [5]
 * Conditional gradient [6] and Generalized conditional gradient for regularized OT [7].
@@ -213,3 +214,5 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 [15] Peyré, G., & Cuturi, M. (2018). [Computational Optimal Transport](https://arxiv.org/pdf/1803.00567.pdf) .
 
 [16] Agueh, M., & Carlier, G. (2011). [Barycenters in the Wasserstein space](https://hal.archives-ouvertes.fr/hal-00637399/document). SIAM Journal on Mathematical Analysis, 43(2), 904-924.
+
+[17] Blondel, M., Seguy, V., & Rolet, A. (2018). [Smooth and Sparse Optimal Transport](https://arxiv.org/abs/1710.06276). Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics (AISTATS).

examples/plot_OT_1D_smooth.py

@@ -0,0 +1,110 @@
+# -*- coding: utf-8 -*-
+"""
+===========================
+1D smooth optimal transport
+===========================
+
+This example illustrates the computation of EMD, Sinkhorn and smooth OT plans
+and their visualization.
+
+"""
+
+# 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 get_1D_gauss as gauss
+
+##############################################################################
+# Generate data
+# -------------
+
+
+#%% 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
+# ----------------------------------
+
+#%% 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')
+
+##############################################################################
+# Solve EMD
+# ---------
+
+
+#%% EMD
+
+G0 = ot.emd(a, b, M)
+
+pl.figure(3, figsize=(5, 5))
+ot.plot.plot1D_mat(a, b, G0, 'OT matrix G0')
+
+##############################################################################
+# Solve Sinkhorn
+# --------------
+
+
+#%% 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()
+
+##############################################################################
+# Solve Smooth OT
+# --------------
+
+
+#%% 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()

ot/__init__.py

@@ -18,6 +18,8 @@
 from . import datasets
 from . import da
 from . import gromov
+from . import smooth
+
 
 # OT functions
 from .lp import emd, emd2