[MRG] Gromov-Wasserstein closed form for linesearch and integration of Fused Gromov-Wasserstein

README.md

@@ -164,6 +164,7 @@ The contributors to this library are:
 * Erwan Vautier (Gromov-Wasserstein)
 * [Kilian Fatras](https://kilianfatras.github.io/)
 * [Alain Rakotomamonjy](https://sites.google.com/site/alainrakotomamonjy/home)
+* [Vayer Titouan](https://tvayer.github.io/)
 
 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):
 
@@ -233,3 +234,5 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 [22] J. Altschuler, J.Weed, P. Rigollet, (2017) [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://papers.nips.cc/paper/6792-near-linear-time-approximation-algorithms-for-optimal-transport-via-sinkhorn-iteration.pdf), Advances in Neural Information Processing Systems (NIPS) 31
 
 [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).

examples/plot_barycenter_fgw.py

@@ -0,0 +1,184 @@
+# -*- coding: utf-8 -*-
+"""
+=================================
+Plot graphs' barycenter using FGW
+=================================
+
+This example illustrates the computation barycenter of labeled graphs using FGW
+
+Requires networkx >=2
+
+.. [18] Vayer Titouan, Chapel Laetitia, Flamary R{\'e}mi, Tavenard Romain
+      and Courty Nicolas
+    "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
+
+#%% load libraries
+import numpy as np
+import matplotlib.pyplot as plt
+import networkx as nx
+import math
+from scipy.sparse.csgraph import shortest_path
+import matplotlib.colors as mcol
+from matplotlib import cm
+from ot.gromov import fgw_barycenters
+#%% Graph functions
+
+
+def find_thresh(C, inf=0.5, sup=3, step=10):
+    """ Trick to find the adequate thresholds from where value of the C matrix are considered close enough to say that nodes are connected
+        Tthe threshold is found by a linesearch between values "inf" and "sup" with "step" thresholds tested.
+        The optimal threshold is the one which minimizes the reconstruction error between the shortest_path matrix coming from the thresholded adjency matrix
+        and the original matrix.
+    Parameters
+    ----------
+    C : ndarray, shape (n_nodes,n_nodes)
+            The structure matrix to threshold
+    inf : float
+          The beginning of the linesearch
+    sup : float
+          The end of the linesearch
+    step : integer
+            Number of thresholds tested
+    """
+    dist = []
+    search = np.linspace(inf, sup, step)
+    for thresh in search:
+        Cprime = sp_to_adjency(C, 0, thresh)
+        SC = shortest_path(Cprime, method='D')
+        SC[SC == float('inf')] = 100
+        dist.append(np.linalg.norm(SC - C))
+    return search[np.argmin(dist)], dist
+
+
+def sp_to_adjency(C, threshinf=0.2, threshsup=1.8):
+    """ Thresholds the structure matrix in order to compute an adjency matrix.
+    All values between threshinf and threshsup are considered representing connected nodes and set to 1. Else are set to 0
+    Parameters
+    ----------
+    C : ndarray, shape (n_nodes,n_nodes)
+        The structure matrix to threshold
+    threshinf : float
+        The minimum value of distance from which the new value is set to 1
+    threshsup : float
+        The maximum value of distance from which the new value is set to 1
+    Returns
+    -------
+    C : ndarray, shape (n_nodes,n_nodes)
+        The threshold matrix. Each element is in {0,1}
+    """
+    H = np.zeros_like(C)
+    np.fill_diagonal(H, np.diagonal(C))
+    C = C - H
+    C = np.minimum(np.maximum(C, threshinf), threshsup)
+    C[C == threshsup] = 0
+    C[C != 0] = 1
+
+    return C
+
+
+def build_noisy_circular_graph(N=20, mu=0, sigma=0.3, with_noise=False, structure_noise=False, p=None):
+    """ Create a noisy circular graph
+    """
+    g = nx.Graph()
+    g.add_nodes_from(list(range(N)))
+    for i in range(N):
+        noise = float(np.random.normal(mu, sigma, 1))
+        if with_noise:
+            g.add_node(i, attr_name=math.sin((2 * i * math.pi / N)) + noise)
+        else:
+            g.add_node(i, attr_name=math.sin(2 * i * math.pi / N))
+        g.add_edge(i, i + 1)
+        if structure_noise:
+            randomint = np.random.randint(0, p)
+            if randomint == 0:
+                if i <= N - 3:
+                    g.add_edge(i, i + 2)
+                if i == N - 2:
+                    g.add_edge(i, 0)
+                if i == N - 1:
+                    g.add_edge(i, 1)
+    g.add_edge(N, 0)
+    noise = float(np.random.normal(mu, sigma, 1))
+    if with_noise:
+        g.add_node(N, attr_name=math.sin((2 * N * math.pi / N)) + noise)
+    else:
+        g.add_node(N, attr_name=math.sin(2 * N * math.pi / N))
+    return g
+
+
+def graph_colors(nx_graph, vmin=0, vmax=7):
+    cnorm = mcol.Normalize(vmin=vmin, vmax=vmax)
+    cpick = cm.ScalarMappable(norm=cnorm, cmap='viridis')
+    cpick.set_array([])
+    val_map = {}
+    for k, v in nx.get_node_attributes(nx_graph, 'attr_name').items():
+        val_map[k] = cpick.to_rgba(v)
+    colors = []
+    for node in nx_graph.nodes():
+        colors.append(val_map[node])
+    return colors
+
+##############################################################################
+# Generate data
+# -------------
+
+#%% circular dataset
+# We build a dataset of noisy circular graphs.
+# Noise is added on the structures by random connections and on the features by gaussian noise.
+
+
+np.random.seed(30)
+X0 = []
+for k in range(9):
+    X0.append(build_noisy_circular_graph(np.random.randint(15, 25), with_noise=True, structure_noise=True, p=3))
+
+##############################################################################
+# Plot data
+# ---------
+
+#%% Plot graphs
+
+plt.figure(figsize=(8, 10))
+for i in range(len(X0)):
+    plt.subplot(3, 3, i + 1)
+    g = X0[i]
+    pos = nx.kamada_kawai_layout(g)
+    nx.draw(g, pos=pos, node_color=graph_colors(g, vmin=-1, vmax=1), with_labels=False, node_size=100)
+plt.suptitle('Dataset of noisy graphs. Color indicates the label', fontsize=20)
+plt.show()
+
+##############################################################################
+# Barycenter computation
+# ----------------------
+
+#%% We compute the barycenter using FGW. Structure matrices are computed using the shortest_path distance in the graph
+# Features distances are the euclidean distances
+Cs = [shortest_path(nx.adjacency_matrix(x)) for x in X0]
+ps = [np.ones(len(x.nodes())) / len(x.nodes()) for x in X0]
+Ys = [np.array([v for (k, v) in nx.get_node_attributes(x, 'attr_name').items()]).reshape(-1, 1) for x in X0]
+lambdas = np.array([np.ones(len(Ys)) / len(Ys)]).ravel()
+sizebary = 15  # we choose a barycenter with 15 nodes
+
+A, C, log = fgw_barycenters(sizebary, Ys, Cs, ps, lambdas, alpha=0.95)
+
+##############################################################################
+# Plot Barycenter
+# -------------------------
+
+#%% Create the barycenter
+bary = nx.from_numpy_matrix(sp_to_adjency(C, threshinf=0, threshsup=find_thresh(C, sup=100, step=100)[0]))
+for i, v in enumerate(A.ravel()):
+    bary.add_node(i, attr_name=v)
+
+#%%
+pos = nx.kamada_kawai_layout(bary)
+nx.draw(bary, pos=pos, node_color=graph_colors(bary, vmin=-1, vmax=1), with_labels=False)
+plt.suptitle('Barycenter', fontsize=20)
+plt.show()

examples/plot_fgw.py

@@ -0,0 +1,173 @@
+# -*- coding: utf-8 -*-
+"""
+==============================
+Plot Fused-gromov-Wasserstein
+==============================
+
+This example illustrates the computation of FGW for 1D measures[18].
+
+.. [18] Vayer Titouan, Chapel Laetitia, Flamary R{\'e}mi, Tavenard Romain
+      and Courty Nicolas
+    "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
+
+##############################################################################
+# Generate data
+# ---------
+
+#%% parameters
+# 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)
+
+##############################################################################
+# Plot data
+# ---------
+
+#%% plot the distributions
+
+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, usetex=True, 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, usetex=True, y=1)
+pl.yticks(())
+pl.tight_layout()
+pl.show()
+
+##############################################################################
+# Create structure matrices and across-feature distance matrix
+# ---------
+
+#%% Structure matrices and across-features distance matrix
+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)
+
+##############################################################################
+# Plot matrices
+# ---------
+
+#%%
+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()
+
+##############################################################################
+# Compute FGW/GW
+# ---------
+
+#%% Computing FGW and GW
+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)
+
+##############################################################################
+# Visualize transport matrices
+# ---------
+
+#%% visu OT matrix
+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()

ot/bregman.py

@@ -6,6 +6,7 @@
 # Author: Remi Flamary <remi.flamary@unice.fr>
 #         Nicolas Courty <ncourty@irisa.fr>
 #         Kilian Fatras <kilian.fatras@irisa.fr>
+#         Titouan Vayer <titouan.vayer@irisa.fr>
 #
 # License: MIT License