CircleCI update of dev docs (2804).

Author: Circle CI

master/_downloads/0228827bdcc2a6735b1faf26c864145d/plot_OT_1D_smooth.zip

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+# -*- coding: utf-8 -*-
+
+r"""
+=====================================================
+Semi-relaxed (Fused) Gromov-Wasserstein Barycenter as Dictionary Learning
+=====================================================
+
+In this example, we illustrate how to learn a semi-relaxed Gromov-Wasserstein
+(srGW) barycenter using a Block-Coordinate Descent algorithm, on a dataset of
+structured data such as graphs, denoted :math:`\{ \mathbf{C_s} \}_{s \in [S]}`
+where every nodes have uniform weights :math:`\{ \mathbf{p_s} \}_{s \in [S]}`.
+Given a barycenter structure matrix :math:`\mathbf{C}` with N nodes,
+each graph :math:`(\mathbf{C_s}, \mathbf{p_s})` is modeled as a reweighed subgraph
+with structure :math:`\mathbf{C}` and weights :math:`\mathbf{w_s} \in \Sigma_N`
+where each :math:`\mathbf{w_s}` corresponds to the second marginal of the OT
+:math:`\mathbf{T_s}` (s.t :math:`\mathbf{w_s} = \mathbf{T_s}^\top \mathbf{1}`)
+minimizing the srGW loss between the s^{th} input and the barycenter.
+
+
+First, we consider a dataset composed of graphs generated by Stochastic Block models
+with variable sizes taken in :math:`\{30, ... , 50\}` and number of clusters
+varying in :math:`\{ 1, 2, 3\}` with random proportions. We learn a srGW barycenter
+with 3 nodes and visualize the learned structure and the embeddings for some inputs.
+
+Second, we illustrate the extension of this framework to graphs endowed
+with node features by using the semi-relaxed Fused Gromov-Wasserstein
+divergence (srFGW). Starting from the aforementioned dataset of unattributed graphs, we
+add discrete labels uniformly depending on the number of clusters. Then conduct
+the analog analysis.
+
+
+[48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty.
+"Semi-relaxed Gromov-Wasserstein divergence and applications on graphs".
+International Conference on Learning Representations (ICLR), 2022.
+
+"""
+# Author: Cédric Vincent-Cuaz <[email protected]>
+#
+# License: MIT License
+
+# sphinx_gallery_thumbnail_number = 2
+
+import numpy as np
+import matplotlib.pylab as pl
+from sklearn.manifold import MDS
+from ot.gromov import (
+    semirelaxed_gromov_barycenters, semirelaxed_fgw_barycenters)
+import ot
+import networkx
+from networkx.generators.community import stochastic_block_model as sbm
+
+#############################################################################
+#
+# Generate a dataset composed of graphs following Stochastic Block models of 1, 2 and 3 clusters.
+# -----------------------------------------------------------------------------------------------
+
+np.random.seed(42)
+
+n_samples = 60  # number of graphs in the dataset
+# For every number of clusters, we generate SBM with fixed inter/intra-clusters probability,
+# and variable cluster proportions.
+clusters = [1, 2, 3]
+Nc = n_samples // len(clusters)  # number of graphs by cluster
+nlabels = len(clusters)
+dataset = []
+node_labels = []
+labels = []
+
+p_inter = 0.1
+p_intra = 0.9
+for n_cluster in clusters:
+    for i in range(Nc):
+        n_nodes = int(np.random.uniform(low=30, high=50))
+
+        if n_cluster > 1:
+            P = p_inter * np.ones((n_cluster, n_cluster))
+            np.fill_diagonal(P, p_intra)
+            props = np.random.uniform(0.2, 1, size=(n_cluster,))
+            props /= props.sum()
+            sizes = np.round(n_nodes * props).astype(np.int32)
+        else:
+            P = p_intra * np.eye(1)
+            sizes = [n_nodes]
+
+        G = sbm(sizes, P, seed=i, directed=False)
+        part = np.array([G.nodes[i]['block'] for i in range(np.sum(sizes))])
+        C = networkx.to_numpy_array(G)
+        dataset.append(C)
+        node_labels.append(part)
+        labels.append(n_cluster)
+
+
+# Visualize samples
+
+def plot_graph(x, C, binary=True, color='C0', s=None):
+    for j in range(C.shape[0]):
+        for i in range(j):
+            if binary:
+                if C[i, j] > 0:
+                    pl.plot([x[i, 0], x[j, 0]], [x[i, 1], x[j, 1]], alpha=0.2, color='k')
+            else:  # connection intensity proportional to C[i,j]
+                pl.plot([x[i, 0], x[j, 0]], [x[i, 1], x[j, 1]], alpha=C[i, j], color='k')
+
+    pl.scatter(x[:, 0], x[:, 1], c=color, s=s, zorder=10, edgecolors='k', cmap='tab10', vmax=9)
+
+
+pl.figure(1, (12, 8))
+pl.clf()
+for idx_c, c in enumerate(clusters):
+    C = dataset[(c - 1) * Nc]  # sample with c clusters
+    # get 2d position for nodes
+    x = MDS(dissimilarity='precomputed', random_state=0).fit_transform(1 - C)
+    pl.subplot(2, nlabels, c)
+    pl.title('(graph) sample from label ' + str(c), fontsize=14)
+    plot_graph(x, C, binary=True, color='C0', s=50.)
+    pl.axis("off")
+    pl.subplot(2, nlabels, nlabels + c)
+    pl.title('(matrix) sample from label %s \n' % c, fontsize=14)
+    pl.imshow(C, interpolation='nearest')
+    pl.axis("off")
+pl.tight_layout()
+pl.show()
+
+#############################################################################
+#
+# Estimate the srGW barycenter from the dataset and visualize embeddings
+# -----------------------------------------------------------
+
+
+np.random.seed(0)
+ps = [ot.unif(C.shape[0]) for C in dataset]  # uniform weights on input nodes
+lambdas = [1. / n_samples for _ in range(n_samples)]  # uniform barycenter
+N = 3  # 3 nodes in the barycenter
+
+# Here we use the Fluid partitioning method to deduce initial transport plans
+# for the barycenter problem. An initlal structure is also deduced from these
+# initial transport plans. Then a warmstart strategy is used iteratively to
+# init each individual srGW problem within the BCD algorithm.
+
+init_plan = 'fluid'  # notice that several init options are implemented in `ot.gromov.semirelaxed_init_plan`
+warmstartT = True
+
+C, log = semirelaxed_gromov_barycenters(
+    N=N, Cs=dataset, ps=ps, lambdas=lambdas, loss_fun='square_loss',
+    tol=1e-6, stop_criterion='loss', warmstartT=warmstartT, log=True,
+    G0=init_plan, verbose=False)
+
+print('barycenter structure:', C)
+
+unmixings = log['p']
+# Compute the 2D representation of the embeddings living in the 2-simplex of probability
+unmixings2D = np.zeros(shape=(n_samples, 2))
+for i, w in enumerate(unmixings):
+    unmixings2D[i, 0] = (2. * w[1] + w[2]) / 2.
+    unmixings2D[i, 1] = (np.sqrt(3.) * w[2]) / 2.
+x = [0., 0.]
+y = [1., 0.]
+z = [0.5, np.sqrt(3) / 2.]
+extremities = np.stack([x, y, z])
+
+pl.figure(2, (4, 4))
+pl.clf()
+pl.title('Embedding space', fontsize=14)
+for cluster in range(nlabels):
+    start, end = Nc * cluster, Nc * (cluster + 1)
+    if cluster == 0:
+        pl.scatter(unmixings2D[start:end, 0], unmixings2D[start:end, 1], c='C' + str(cluster), marker='o', s=80., label='1 cluster')
+    else:
+        pl.scatter(unmixings2D[start:end, 0], unmixings2D[start:end, 1], c='C' + str(cluster), marker='o', s=80., label='%s clusters' % (cluster + 1))
+pl.scatter(extremities[:, 0], extremities[:, 1], c='black', marker='x', s=100., label='bary. nodes')
+pl.plot([x[0], y[0]], [x[1], y[1]], color='black', linewidth=2.)
+pl.plot([x[0], z[0]], [x[1], z[1]], color='black', linewidth=2.)
+pl.plot([y[0], z[0]], [y[1], z[1]], color='black', linewidth=2.)
+pl.axis('off')
+pl.legend(fontsize=11)
+pl.tight_layout()
+pl.show()
+
+#############################################################################
+#
+# Endow the dataset with node features
+# ------------------------------------
+# node labels, corresponding to the true SBM cluster assignments,
+# are set for each graph as one-hot encoded node features.
+
+dataset_features = []
+for i in range(len(dataset)):
+    n = dataset[i].shape[0]
+    F = np.zeros((n, 3))
+    F[np.arange(n), node_labels[i]] = 1.
+    dataset_features.append(F)
+
+pl.figure(3, (12, 8))
+pl.clf()
+for idx_c, c in enumerate(clusters):
+    C = dataset[(c - 1) * Nc]  # sample with c clusters
+    F = dataset_features[(c - 1) * Nc]
+    colors = [f'C{labels[i]}' for i in range(F.shape[0])]
+    # get 2d position for nodes
+    x = MDS(dissimilarity='precomputed', random_state=0).fit_transform(1 - C)
+    pl.subplot(2, nlabels, c)
+    pl.title('(graph) sample from label ' + str(c), fontsize=14)
+    plot_graph(x, C, binary=True, color=colors, s=50)
+    pl.axis("off")
+    pl.subplot(2, nlabels, nlabels + c)
+    pl.title('(matrix) sample from label %s \n' % c, fontsize=14)
+    pl.imshow(C, interpolation='nearest')
+    pl.axis("off")
+pl.tight_layout()
+pl.show()
+
+#############################################################################
+#
+# Estimate the srFGW barycenter from the attributed graphs and visualize embeddings
+# -----------------------------------------------------------
+# We emphasize the dependence to the trade-off parameter alpha that weights the
+# relative importance between structures (alpha=1) and features (alpha=0),
+# knowing that embeddings that perfectly cluster graphs w.r.t their features
+# should ease the identification of the number of clusters in the graphs.
+
+list_alphas = [0.0001, 0.5, 0.9999]
+list_unmixings2D = []
+
+for ialpha, alpha in enumerate(list_alphas):
+    print('--- alpha:', alpha)
+    C, F, log = semirelaxed_fgw_barycenters(
+        N=N, Ys=dataset_features, Cs=dataset, ps=ps, lambdas=lambdas,
+        alpha=alpha, loss_fun='square_loss', tol=1e-6, stop_criterion='loss',
+        warmstartT=warmstartT, log=True, G0=init_plan)
+
+    print('barycenter structure:', C)
+    print('barycenter features:', F)
+
+    unmixings = log['p']
+    # Compute the 2D representation of the embeddings living in the 2-simplex of probability
+    unmixings2D = np.zeros(shape=(n_samples, 2))
+    for i, w in enumerate(unmixings):
+        unmixings2D[i, 0] = (2. * w[1] + w[2]) / 2.
+        unmixings2D[i, 1] = (np.sqrt(3.) * w[2]) / 2.
+    list_unmixings2D.append(unmixings2D.copy())
+
+x = [0., 0.]
+y = [1., 0.]
+z = [0.5, np.sqrt(3) / 2.]
+extremities = np.stack([x, y, z])
+
+pl.figure(4, (12, 4))
+pl.clf()
+pl.suptitle('Embedding spaces', fontsize=14)
+for ialpha, alpha in enumerate(list_alphas):
+    pl.subplot(1, len(list_alphas), ialpha + 1)
+    pl.title(f'alpha = {alpha}', fontsize=14)
+    for cluster in range(nlabels):
+        start, end = Nc * cluster, Nc * (cluster + 1)
+        if cluster == 0:
+            pl.scatter(list_unmixings2D[ialpha][start:end, 0], list_unmixings2D[ialpha][start:end, 1], c='C' + str(cluster), marker='o', s=80., label='1 cluster')
+        else:
+            pl.scatter(list_unmixings2D[ialpha][start:end, 0], list_unmixings2D[ialpha][start:end, 1], c='C' + str(cluster), marker='o', s=80., label='%s clusters' % (cluster + 1))
+    pl.scatter(extremities[:, 0], extremities[:, 1], c='black', marker='x', s=100., label='bary. nodes')
+    pl.plot([x[0], y[0]], [x[1], y[1]], color='black', linewidth=2.)
+    pl.plot([x[0], z[0]], [x[1], z[1]], color='black', linewidth=2.)
+    pl.plot([y[0], z[0]], [y[1], z[1]], color='black', linewidth=2.)
+    pl.axis('off')
+    pl.legend(fontsize=11)
+pl.tight_layout()
+pl.show()