5 files changed, 615 insertions, 0 deletions
diff --git a/examples/barycenters/README.txt b/examples/barycenters/README.txt
new file mode 100644
index 0000000..8461f7f
--- /dev/null
+++ b/examples/barycenters/README.txt
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+
+
+Wasserstein barycenters
+-----------------------
+\ No newline at end of file
diff --git a/examples/barycenters/plot_barycenter_1D.py b/examples/barycenters/plot_barycenter_1D.py
new file mode 100644
index 0000000..63dc460
--- /dev/null
+++ b/examples/barycenters/plot_barycenter_1D.py
@@ -0,0 +1,162 @@
+# -*- coding: utf-8 -*-
+"""
+==============================
+1D Wasserstein barycenter demo
+==============================
+
+This example illustrates the computation of regularized Wassersyein Barycenter
+as proposed in [3].
+
+
+[3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015).
+Iterative Bregman projections for regularized transportation problems
+SIAM Journal on Scientific Computing, 37(2), A1111-A1138.
+
+"""
+
+# Author: Remi Flamary <remi.flamary@unice.fr>
+#
+# License: MIT License
+
+# sphinx_gallery_thumbnail_number = 4
+
+import numpy as np
+import matplotlib.pylab as pl
+import ot
+# necessary for 3d plot even if not used
+from mpl_toolkits.mplot3d import Axes3D  # noqa
+from matplotlib.collections import PolyCollection
+
+##############################################################################
+# Generate data
+# -------------
+
+#%% parameters
+
+n = 100  # nb bins
+
+# bin positions
+x = np.arange(n, dtype=np.float64)
+
+# Gaussian distributions
+a1 = ot.datasets.make_1D_gauss(n, m=20, s=5)  # m= mean, s= std
+a2 = ot.datasets.make_1D_gauss(n, m=60, s=8)
+
+# creating matrix A containing all distributions
+A = np.vstack((a1, a2)).T
+n_distributions = A.shape[1]
+
+# loss matrix + normalization
+M = ot.utils.dist0(n)
+M /= M.max()
+
+##############################################################################
+# Plot data
+# ---------
+
+#%% plot the distributions
+
+pl.figure(1, figsize=(6.4, 3))
+for i in range(n_distributions):
+    pl.plot(x, A[:, i])
+pl.title('Distributions')
+pl.tight_layout()
+
+##############################################################################
+# Barycenter computation
+# ----------------------
+
+#%% barycenter computation
+
+alpha = 0.2  # 0<=alpha<=1
+weights = np.array([1 - alpha, alpha])
+
+# l2bary
+bary_l2 = A.dot(weights)
+
+# wasserstein
+reg = 1e-3
+bary_wass = ot.bregman.barycenter(A, M, reg, weights)
+
+pl.figure(2)
+pl.clf()
+pl.subplot(2, 1, 1)
+for i in range(n_distributions):
+    pl.plot(x, A[:, i])
+pl.title('Distributions')
+
+pl.subplot(2, 1, 2)
+pl.plot(x, bary_l2, 'r', label='l2')
+pl.plot(x, bary_wass, 'g', label='Wasserstein')
+pl.legend()
+pl.title('Barycenters')
+pl.tight_layout()
+
+##############################################################################
+# Barycentric interpolation
+# -------------------------
+
+#%% barycenter interpolation
+
+n_alpha = 11
+alpha_list = np.linspace(0, 1, n_alpha)
+
+
+B_l2 = np.zeros((n, n_alpha))
+
+B_wass = np.copy(B_l2)
+
+for i in range(0, n_alpha):
+    alpha = alpha_list[i]
+    weights = np.array([1 - alpha, alpha])
+    B_l2[:, i] = A.dot(weights)
+    B_wass[:, i] = ot.bregman.barycenter(A, M, reg, weights)
+
+#%% plot interpolation
+
+pl.figure(3)
+
+cmap = pl.cm.get_cmap('viridis')
+verts = []
+zs = alpha_list
+for i, z in enumerate(zs):
+    ys = B_l2[:, i]
+    verts.append(list(zip(x, ys)))
+
+ax = pl.gcf().gca(projection='3d')
+
+poly = PolyCollection(verts, facecolors=[cmap(a) for a in alpha_list])
+poly.set_alpha(0.7)
+ax.add_collection3d(poly, zs=zs, zdir='y')
+ax.set_xlabel('x')
+ax.set_xlim3d(0, n)
+ax.set_ylabel('$\\alpha$')
+ax.set_ylim3d(0, 1)
+ax.set_zlabel('')
+ax.set_zlim3d(0, B_l2.max() * 1.01)
+pl.title('Barycenter interpolation with l2')
+pl.tight_layout()
+
+pl.figure(4)
+cmap = pl.cm.get_cmap('viridis')
+verts = []
+zs = alpha_list
+for i, z in enumerate(zs):
+    ys = B_wass[:, i]
+    verts.append(list(zip(x, ys)))
+
+ax = pl.gcf().gca(projection='3d')
+
+poly = PolyCollection(verts, facecolors=[cmap(a) for a in alpha_list])
+poly.set_alpha(0.7)
+ax.add_collection3d(poly, zs=zs, zdir='y')
+ax.set_xlabel('x')
+ax.set_xlim3d(0, n)
+ax.set_ylabel('$\\alpha$')
+ax.set_ylim3d(0, 1)
+ax.set_zlabel('')
+ax.set_zlim3d(0, B_l2.max() * 1.01)
+pl.title('Barycenter interpolation with Wasserstein')
+pl.tight_layout()
+
+pl.show()
diff --git a/examples/barycenters/plot_barycenter_lp_vs_entropic.py b/examples/barycenters/plot_barycenter_lp_vs_entropic.py
new file mode 100644
index 0000000..57a6bac
--- /dev/null
+++ b/examples/barycenters/plot_barycenter_lp_vs_entropic.py
@@ -0,0 +1,288 @@
+# -*- coding: utf-8 -*-
+"""
+=================================================================================
+1D Wasserstein barycenter comparison between exact LP and entropic regularization
+=================================================================================
+
+This example illustrates the computation of regularized Wasserstein Barycenter
+as proposed in [3] and exact LP barycenters using standard LP solver.
+
+It reproduces approximately Figure 3.1 and 3.2 from the following paper:
+Cuturi, M., & Peyré, G. (2016). A smoothed dual approach for variational
+Wasserstein problems. SIAM Journal on Imaging Sciences, 9(1), 320-343.
+
+[3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015).
+Iterative Bregman projections for regularized transportation problems
+SIAM Journal on Scientific Computing, 37(2), A1111-A1138.
+
+"""
+
+# Author: Remi Flamary <remi.flamary@unice.fr>
+#
+# License: MIT License
+
+# sphinx_gallery_thumbnail_number = 4
+
+import numpy as np
+import matplotlib.pylab as pl
+import ot
+# necessary for 3d plot even if not used
+from mpl_toolkits.mplot3d import Axes3D  # noqa
+from matplotlib.collections import PolyCollection  # noqa
+
+#import ot.lp.cvx as cvx
+
+##############################################################################
+# Gaussian Data
+# -------------
+
+#%% parameters
+
+problems = []
+
+n = 100  # nb bins
+
+# bin positions
+x = np.arange(n, dtype=np.float64)
+
+# Gaussian distributions
+# Gaussian distributions
+a1 = ot.datasets.make_1D_gauss(n, m=20, s=5)  # m= mean, s= std
+a2 = ot.datasets.make_1D_gauss(n, m=60, s=8)
+
+# creating matrix A containing all distributions
+A = np.vstack((a1, a2)).T
+n_distributions = A.shape[1]
+
+# loss matrix + normalization
+M = ot.utils.dist0(n)
+M /= M.max()
+
+
+#%% plot the distributions
+
+pl.figure(1, figsize=(6.4, 3))
+for i in range(n_distributions):
+    pl.plot(x, A[:, i])
+pl.title('Distributions')
+pl.tight_layout()
+
+#%% barycenter computation
+
+alpha = 0.5  # 0<=alpha<=1
+weights = np.array([1 - alpha, alpha])
+
+# l2bary
+bary_l2 = A.dot(weights)
+
+# wasserstein
+reg = 1e-3
+ot.tic()
+bary_wass = ot.bregman.barycenter(A, M, reg, weights)
+ot.toc()
+
+
+ot.tic()
+bary_wass2 = ot.lp.barycenter(A, M, weights, solver='interior-point', verbose=True)
+ot.toc()
+
+pl.figure(2)
+pl.clf()
+pl.subplot(2, 1, 1)
+for i in range(n_distributions):
+    pl.plot(x, A[:, i])
+pl.title('Distributions')
+
+pl.subplot(2, 1, 2)
+pl.plot(x, bary_l2, 'r', label='l2')
+pl.plot(x, bary_wass, 'g', label='Reg Wasserstein')
+pl.plot(x, bary_wass2, 'b', label='LP Wasserstein')
+pl.legend()
+pl.title('Barycenters')
+pl.tight_layout()
+
+problems.append([A, [bary_l2, bary_wass, bary_wass2]])
+
+##############################################################################
+# Stair Data
+# ----------
+
+#%% parameters
+
+a1 = 1.0 * (x > 10) * (x < 50)
+a2 = 1.0 * (x > 60) * (x < 80)
+
+a1 /= a1.sum()
+a2 /= a2.sum()
+
+# creating matrix A containing all distributions
+A = np.vstack((a1, a2)).T
+n_distributions = A.shape[1]
+
+# loss matrix + normalization
+M = ot.utils.dist0(n)
+M /= M.max()
+
+
+#%% plot the distributions
+
+pl.figure(1, figsize=(6.4, 3))
+for i in range(n_distributions):
+    pl.plot(x, A[:, i])
+pl.title('Distributions')
+pl.tight_layout()
+
+
+#%% barycenter computation
+
+alpha = 0.5  # 0<=alpha<=1
+weights = np.array([1 - alpha, alpha])
+
+# l2bary
+bary_l2 = A.dot(weights)
+
+# wasserstein
+reg = 1e-3
+ot.tic()
+bary_wass = ot.bregman.barycenter(A, M, reg, weights)
+ot.toc()
+
+
+ot.tic()
+bary_wass2 = ot.lp.barycenter(A, M, weights, solver='interior-point', verbose=True)
+ot.toc()
+
+
+problems.append([A, [bary_l2, bary_wass, bary_wass2]])
+
+pl.figure(2)
+pl.clf()
+pl.subplot(2, 1, 1)
+for i in range(n_distributions):
+    pl.plot(x, A[:, i])
+pl.title('Distributions')
+
+pl.subplot(2, 1, 2)
+pl.plot(x, bary_l2, 'r', label='l2')
+pl.plot(x, bary_wass, 'g', label='Reg Wasserstein')
+pl.plot(x, bary_wass2, 'b', label='LP Wasserstein')
+pl.legend()
+pl.title('Barycenters')
+pl.tight_layout()
+
+
+##############################################################################
+# Dirac Data
+# ----------
+
+#%% parameters
+
+a1 = np.zeros(n)
+a2 = np.zeros(n)
+
+a1[10] = .25
+a1[20] = .5
+a1[30] = .25
+a2[80] = 1
+
+
+a1 /= a1.sum()
+a2 /= a2.sum()
+
+# creating matrix A containing all distributions
+A = np.vstack((a1, a2)).T
+n_distributions = A.shape[1]
+
+# loss matrix + normalization
+M = ot.utils.dist0(n)
+M /= M.max()
+
+
+#%% plot the distributions
+
+pl.figure(1, figsize=(6.4, 3))
+for i in range(n_distributions):
+    pl.plot(x, A[:, i])
+pl.title('Distributions')
+pl.tight_layout()
+
+
+#%% barycenter computation
+
+alpha = 0.5  # 0<=alpha<=1
+weights = np.array([1 - alpha, alpha])
+
+# l2bary
+bary_l2 = A.dot(weights)
+
+# wasserstein
+reg = 1e-3
+ot.tic()
+bary_wass = ot.bregman.barycenter(A, M, reg, weights)
+ot.toc()
+
+
+ot.tic()
+bary_wass2 = ot.lp.barycenter(A, M, weights, solver='interior-point', verbose=True)
+ot.toc()
+
+
+problems.append([A, [bary_l2, bary_wass, bary_wass2]])
+
+pl.figure(2)
+pl.clf()
+pl.subplot(2, 1, 1)
+for i in range(n_distributions):
+    pl.plot(x, A[:, i])
+pl.title('Distributions')
+
+pl.subplot(2, 1, 2)
+pl.plot(x, bary_l2, 'r', label='l2')
+pl.plot(x, bary_wass, 'g', label='Reg Wasserstein')
+pl.plot(x, bary_wass2, 'b', label='LP Wasserstein')
+pl.legend()
+pl.title('Barycenters')
+pl.tight_layout()
+
+
+##############################################################################
+# Final figure
+# ------------
+#
+
+#%% plot
+
+nbm = len(problems)
+nbm2 = (nbm // 2)
+
+
+pl.figure(2, (20, 6))
+pl.clf()
+
+for i in range(nbm):
+
+    A = problems[i][0]
+    bary_l2 = problems[i][1][0]
+    bary_wass = problems[i][1][1]
+    bary_wass2 = problems[i][1][2]
+
+    pl.subplot(2, nbm, 1 + i)
+    for j in range(n_distributions):
+        pl.plot(x, A[:, j])
+    if i == nbm2:
+        pl.title('Distributions')
+    pl.xticks(())
+    pl.yticks(())
+
+    pl.subplot(2, nbm, 1 + i + nbm)
+
+    pl.plot(x, bary_l2, 'r', label='L2 (Euclidean)')
+    pl.plot(x, bary_wass, 'g', label='Reg Wasserstein')
+    pl.plot(x, bary_wass2, 'b', label='LP Wasserstein')
+    if i == nbm - 1:
+        pl.legend()
+    if i == nbm2:
+        pl.title('Barycenters')
+
+    pl.xticks(())
+    pl.yticks(())
diff --git a/examples/barycenters/plot_convolutional_barycenter.py b/examples/barycenters/plot_convolutional_barycenter.py
new file mode 100644
index 0000000..cbcd4a1
--- /dev/null
+++ b/examples/barycenters/plot_convolutional_barycenter.py
@@ -0,0 +1,92 @@
+
+#%%
+# -*- coding: utf-8 -*-
+"""
+============================================
+Convolutional Wasserstein Barycenter example
+============================================
+
+This example is designed to illustrate how the Convolutional Wasserstein Barycenter
+function of POT works.
+"""
+
+# Author: Nicolas Courty <ncourty@irisa.fr>
+#
+# License: MIT License
+
+
+import numpy as np
+import pylab as pl
+import ot
+
+##############################################################################
+# Data preparation
+# ----------------
+#
+# The four distributions are constructed from 4 simple images
+
+
+f1 = 1 - pl.imread('../../data/redcross.png')[:, :, 2]
+f2 = 1 - pl.imread('../../data/duck.png')[:, :, 2]
+f3 = 1 - pl.imread('../../data/heart.png')[:, :, 2]
+f4 = 1 - pl.imread('../../data/tooth.png')[:, :, 2]
+
+A = []
+f1 = f1 / np.sum(f1)
+f2 = f2 / np.sum(f2)
+f3 = f3 / np.sum(f3)
+f4 = f4 / np.sum(f4)
+A.append(f1)
+A.append(f2)
+A.append(f3)
+A.append(f4)
+A = np.array(A)
+
+nb_images = 5
+
+# those are the four corners coordinates that will be interpolated by bilinear
+# interpolation
+v1 = np.array((1, 0, 0, 0))
+v2 = np.array((0, 1, 0, 0))
+v3 = np.array((0, 0, 1, 0))
+v4 = np.array((0, 0, 0, 1))
+
+
+##############################################################################
+# Barycenter computation and visualization
+# ----------------------------------------
+#
+
+pl.figure(figsize=(10, 10))
+pl.title('Convolutional Wasserstein Barycenters in POT')
+cm = 'Blues'
+# regularization parameter
+reg = 0.004
+for i in range(nb_images):
+    for j in range(nb_images):
+        pl.subplot(nb_images, nb_images, i * nb_images + j + 1)
+        tx = float(i) / (nb_images - 1)
+        ty = float(j) / (nb_images - 1)
+
+        # weights are constructed by bilinear interpolation
+        tmp1 = (1 - tx) * v1 + tx * v2
+        tmp2 = (1 - tx) * v3 + tx * v4
+        weights = (1 - ty) * tmp1 + ty * tmp2
+
+        if i == 0 and j == 0:
+            pl.imshow(f1, cmap=cm)
+            pl.axis('off')
+        elif i == 0 and j == (nb_images - 1):
+            pl.imshow(f3, cmap=cm)
+            pl.axis('off')
+        elif i == (nb_images - 1) and j == 0:
+            pl.imshow(f2, cmap=cm)
+            pl.axis('off')
+        elif i == (nb_images - 1) and j == (nb_images - 1):
+            pl.imshow(f4, cmap=cm)
+            pl.axis('off')
+        else:
+            # call to barycenter computation
+            pl.imshow(ot.bregman.convolutional_barycenter2d(A, reg, weights), cmap=cm)
+            pl.axis('off')
+pl.show()
diff --git a/examples/barycenters/plot_free_support_barycenter.py b/examples/barycenters/plot_free_support_barycenter.py
new file mode 100644
index 0000000..27ddc8e
--- /dev/null
+++ b/examples/barycenters/plot_free_support_barycenter.py
@@ -0,0 +1,69 @@
+# -*- coding: utf-8 -*-
+"""
+====================================================
+2D free support Wasserstein barycenters of distributions
+====================================================
+
+Illustration of 2D Wasserstein barycenters if distributions are weighted
+sum of diracs.
+
+"""
+
+# Author: Vivien Seguy <vivien.seguy@iip.ist.i.kyoto-u.ac.jp>
+#
+# License: MIT License
+
+import numpy as np
+import matplotlib.pylab as pl
+import ot
+
+
+##############################################################################
+# Generate data
+# -------------
+
+N = 3
+d = 2
+measures_locations = []
+measures_weights = []
+
+for i in range(N):
+
+    n_i = np.random.randint(low=1, high=20)  # nb samples
+
+    mu_i = np.random.normal(0., 4., (d,))  # Gaussian mean
+
+    A_i = np.random.rand(d, d)
+    cov_i = np.dot(A_i, A_i.transpose())  # Gaussian covariance matrix
+
+    x_i = ot.datasets.make_2D_samples_gauss(n_i, mu_i, cov_i)  # Dirac locations
+    b_i = np.random.uniform(0., 1., (n_i,))
+    b_i = b_i / np.sum(b_i)  # Dirac weights
+
+    measures_locations.append(x_i)
+    measures_weights.append(b_i)
+
+
+##############################################################################
+# Compute free support barycenter
+# -------------------------------
+
+k = 10  # number of Diracs of the barycenter
+X_init = np.random.normal(0., 1., (k, d))  # initial Dirac locations
+b = np.ones((k,)) / k  # weights of the barycenter (it will not be optimized, only the locations are optimized)
+
+X = ot.lp.free_support_barycenter(measures_locations, measures_weights, X_init, b)
+
+
+##############################################################################
+# Plot data
+# ---------
+
+pl.figure(1)
+for (x_i, b_i) in zip(measures_locations, measures_weights):
+    color = np.random.randint(low=1, high=10 * N)
+    pl.scatter(x_i[:, 0], x_i[:, 1], s=b_i * 1000, label='input measure')
+pl.scatter(X[:, 0], X[:, 1], s=b * 1000, c='black', marker='^', label='2-Wasserstein barycenter')
+pl.title('Data measures and their barycenter')
+pl.legend(loc=0)
+pl.show()