[MRG] Make gromov loss differentiable wrt matrices and weights (#302)

* grmov differentable

* new stuff

* test gromov gradients

* fgwdifferentiable

* fgw tested

* correc name test

* add awesome example with gromov optimizatrion

* pep8+ typos

* damn pep8

* thunbnail

* remove prints

README.md

@@ -26,7 +26,7 @@ POT provides the following generic OT solvers (links to examples):
 * Debiased Sinkhorn barycenters [Sinkhorn divergence barycenter](https://pythonot.github.io/auto_examples/barycenters/plot_debiased_barycenter.html) [37]
 * [Smooth optimal transport solvers](https://pythonot.github.io/auto_examples/plot_OT_1D_smooth.html) (dual and semi-dual) for KL and squared L2 regularizations [17].
 * Non regularized [Wasserstein barycenters [16] ](https://pythonot.github.io/auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html)) with LP solver (only small scale).
-* [Gromov-Wasserstein distances](https://pythonot.github.io/auto_examples/gromov/plot_gromov.html) and [GW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_gromov_barycenter.html)  (exact [13] and regularized [12])
+* [Gromov-Wasserstein distances](https://pythonot.github.io/auto_examples/gromov/plot_gromov.html) and [GW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_gromov_barycenter.html)  (exact [13] and regularized [12]), differentiable using gradients from
  * [Fused-Gromov-Wasserstein distances solver](https://pythonot.github.io/auto_examples/gromov/plot_fgw.html#sphx-glr-auto-examples-plot-fgw-py) and [FGW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_barycenter_fgw.html) [24]
 * [Stochastic solver](https://pythonot.github.io/auto_examples/plot_stochastic.html) for Large-scale Optimal Transport (semi-dual problem [18] and dual problem [19])
 * [Stochastic solver of Gromov Wasserstein](https://pythonot.github.io/auto_examples/gromov/plot_gromov.html) for large-scale problem with any loss functions [33]
@@ -295,5 +295,8 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 via optimal transport and diffusions](http://proceedings.mlr.press/v97/liutkus19a/liutkus19a.pdf). In International Conference on 
 Machine Learning (pp. 4104-4113). PMLR.
 
-[37] Janati, H., Cuturi, M., Gramfort, A. Proceedings of the 37th International
-Conference on Machine Learning, PMLR 119:4692-4701, 2020
+[37] Janati, H., Cuturi, M., Gramfort, A. [Debiased sinkhorn barycenters](http://proceedings.mlr.press/v119/janati20a/janati20a.pdf) Proceedings of the 37th International
+Conference on Machine Learning, PMLR 119:4692-4701, 2020
+
+[38] C. Vincent-Cuaz, T. Vayer, R. Flamary, M. Corneli, N. Courty, [Online Graph 
+Dictionary Learning](https://arxiv.org/pdf/2102.06555.pdf), International Conference on Machine Learning (ICML), 2021.

examples/backends/plot_optim_gromov_pytorch.py

@@ -0,0 +1,260 @@
+r"""
+=================================
+Optimizing the Gromov-Wasserstein distance with PyTorch
+=================================
+
+In this exemple we use the pytorch backend to optimize the  Gromov-Wasserstein
+(GW) loss between two graphs expressed as empirical distribution.
+
+In the first example we optimize the weights on the node of a simple template
+graph so that it minimizes the GW with a given Stochastic Block Model graph.
+We can see that this actually recovers the proportion of classes in the SBM
+and allows for an accurate clustering of the nodes using the GW optimal plan.
+
+In a second example we optimize simultaneously the weights and the sructure of
+the template graph which allows us to perform graph compression and to recover
+other properties of the SBM.
+
+The backend actually uses the gradients expressed in [38] to optimize the
+weights.
+
+[38] C. Vincent-Cuaz, T. Vayer, R. Flamary, M. Corneli, N. Courty, Online Graph
+Dictionary Learning, International Conference on Machine Learning (ICML), 2021.
+
+"""
+# Author: Rémi Flamary <remi.flamary@polytechnique.edu>
+#
+# License: MIT License
+
+# sphinx_gallery_thumbnail_number = 3
+
+from sklearn.manifold import MDS
+import numpy as np
+import matplotlib.pylab as pl
+import torch
+
+import ot
+from ot.gromov import gromov_wasserstein2
+
+# %%
+# Graph generation
+# ---------------
+
+rng = np.random.RandomState(42)
+
+
+def get_sbm(n, nc, ratio, P):
+    nbpc = np.round(n * ratio).astype(int)
+    n = np.sum(nbpc)
+    C = np.zeros((n, n))
+    for c1 in range(nc):
+        for c2 in range(c1 + 1):
+            if c1 == c2:
+                for i in range(np.sum(nbpc[:c1]), np.sum(nbpc[:c1 + 1])):
+                    for j in range(np.sum(nbpc[:c2]), i):
+                        if rng.rand() <= P[c1, c2]:
+                            C[i, j] = 1
+            else:
+                for i in range(np.sum(nbpc[:c1]), np.sum(nbpc[:c1 + 1])):
+                    for j in range(np.sum(nbpc[:c2]), np.sum(nbpc[:c2 + 1])):
+                        if rng.rand() <= P[c1, c2]:
+                            C[i, j] = 1
+
+    return C + C.T
+
+
+n = 100
+nc = 3
+ratio = np.array([.5, .3, .2])
+P = np.array(0.6 * np.eye(3) + 0.05 * np.ones((3, 3)))
+C1 = get_sbm(n, nc, ratio, P)
+
+# get 2d position for nodes
+x1 = MDS(dissimilarity='precomputed', random_state=0).fit_transform(1 - C1)
+
+
+def plot_graph(x, C, color='C0', s=None):
+    for j in range(C.shape[0]):
+        for i in range(j):
+            if C[i, j] > 0:
+                pl.plot([x[i, 0], x[j, 0]], [x[i, 1], x[j, 1]], alpha=0.2, color='k')
+    pl.scatter(x[:, 0], x[:, 1], c=color, s=s, zorder=10, edgecolors='k', cmap='tab10', vmax=9)
+
+
+pl.figure(1, (10, 5))
+pl.clf()
+pl.subplot(1, 2, 1)
+plot_graph(x1, C1, color='C0')
+pl.title("SBM Graph")
+pl.axis("off")
+pl.subplot(1, 2, 2)
+pl.imshow(C1, interpolation='nearest')
+pl.title("Adjacency matrix")
+pl.axis("off")
+
+
+# %%
+# Optimizing the weights of a simple template C0=eye(3) to fit Graph 1
+# ------------------------------------------------
+# The adajacency matrix C1 is block diagonal with 3 blocks. We want to
+# optimize the weights of a simple template C0=eye(3) and see if we can
+# recover the proportion of classes from the SBM (up to a permutation).
+
+C0 = np.eye(3)
+
+
+def min_weight_gw(C1, C2, a2, nb_iter_max=100, lr=1e-2):
+    """ solve min_a GW(C1,C2,a, a2) by gradient descent"""
+
+    # use pyTorch for our data
+    C1_torch = torch.tensor(C1)
+    C2_torch = torch.tensor(C2)
+
+    a0 = rng.rand(C1.shape[0])  # random_init
+    a0 /= a0.sum()  # on simplex
+    a1_torch = torch.tensor(a0).requires_grad_(True)
+    a2_torch = torch.tensor(a2)
+
+    loss_iter = []
+
+    for i in range(nb_iter_max):
+
+        loss = gromov_wasserstein2(C1_torch, C2_torch, a1_torch, a2_torch)
+
+        loss_iter.append(loss.clone().detach().cpu().numpy())
+        loss.backward()
+
+        #print("{:03d} | {}".format(i, loss_iter[-1]))
+
+        # performs a step of projected gradient descent
+        with torch.no_grad():
+            grad = a1_torch.grad
+            a1_torch -= grad * lr   # step
+            a1_torch.grad.zero_()
+            a1_torch.data = ot.utils.proj_simplex(a1_torch)
+
+    a1 = a1_torch.clone().detach().cpu().numpy()
+
+    return a1, loss_iter
+
+
+a0_est, loss_iter0 = min_weight_gw(C0, C1, ot.unif(n), nb_iter_max=100, lr=1e-2)
+
+pl.figure(2)
+pl.plot(loss_iter0)
+pl.title("Loss along iterations")
+
+print("Estimated weights : ", a0_est)
+print("True proportions : ", ratio)
+
+
+# %%
+# It is clear that the optimization has converged and that we recover the
+# ratio of the different classes in the SBM graph up to a permutation.
+
+
+# %%
+# Community clustering with uniform and estimated weights
+# --------------------------------------------
+# The GW OT  plan can be used to perform a clustering of the nodes of a graph
+# when computing the GW with a simple template like C0 by labeling nodes in
+# the original graph using by the index of the noe in the template receiving
+# the most mass.
+#
+# We show here the result of such a clustering when using uniform weights on
+# the template C0 and when using the optimal weights previously estimated.
+
+
+T_unif = ot.gromov_wasserstein(C1, C0, ot.unif(n), ot.unif(3))
+label_unif = T_unif.argmax(1)
+
+T_est = ot.gromov_wasserstein(C1, C0, ot.unif(n), a0_est)
+label_est = T_est.argmax(1)
+
+pl.figure(3, (10, 5))
+pl.clf()
+pl.subplot(1, 2, 1)
+plot_graph(x1, C1, color=label_unif)
+pl.title("Graph clustering unif. weights")
+pl.axis("off")
+pl.subplot(1, 2, 2)
+plot_graph(x1, C1, color=label_est)
+pl.title("Graph clustering est. weights")
+pl.axis("off")
+
+
+# %%
+# Graph compression with GW
+# -------------------------
+
+# Now we  optimize both the weights and structure of a small graph that
+# minimize the GW distance wrt our data graph. This can be seen as graph
+# compression but can also recover important properties of an SBM such
+# as its class proportion but also its matrix of probability of links between
+# classes
+
+
+def graph_compession_gw(nb_nodes, C2, a2, nb_iter_max=100, lr=1e-2):
+    """ solve min_a GW(C1,C2,a, a2) by gradient descent"""
+
+    # use pyTorch for our data
+
+    C2_torch = torch.tensor(C2)
+    a2_torch = torch.tensor(a2)
+
+    a0 = rng.rand(nb_nodes)  # random_init
+    a0 /= a0.sum()  # on simplex
+    a1_torch = torch.tensor(a0).requires_grad_(True)
+    C0 = np.eye(nb_nodes)
+    C1_torch = torch.tensor(C0).requires_grad_(True)
+
+    loss_iter = []
+
+    for i in range(nb_iter_max):
+
+        loss = gromov_wasserstein2(C1_torch, C2_torch, a1_torch, a2_torch)
+
+        loss_iter.append(loss.clone().detach().cpu().numpy())
+        loss.backward()
+
+        #print("{:03d} | {}".format(i, loss_iter[-1]))
+
+        # performs a step of projected gradient descent
+        with torch.no_grad():
+            grad = a1_torch.grad
+            a1_torch -= grad * lr   # step
+            a1_torch.grad.zero_()
+            a1_torch.data = ot.utils.proj_simplex(a1_torch)
+
+            grad = C1_torch.grad
+            C1_torch -= grad * lr   # step
+            C1_torch.grad.zero_()
+            C1_torch.data = torch.clamp(C1_torch, 0, 1)
+
+    a1 = a1_torch.clone().detach().cpu().numpy()
+    C1 = C1_torch.clone().detach().cpu().numpy()
+
+    return a1, C1, loss_iter
+
+
+nb_nodes = 3
+a0_est2, C0_est2, loss_iter2 = graph_compession_gw(nb_nodes, C1, ot.unif(n),
+                                                   nb_iter_max=100, lr=5e-2)
+
+pl.figure(4)
+pl.plot(loss_iter2)
+pl.title("Loss along iterations")
+
+
+print("Estimated weights : ", a0_est2)
+print("True proportions : ", ratio)
+
+pl.figure(6, (10, 3.5))
+pl.clf()
+pl.subplot(1, 2, 1)
+pl.imshow(P, vmin=0, vmax=1)
+pl.title('True SBM P matrix')
+pl.subplot(1, 2, 2)
+pl.imshow(C0_est2, vmin=0, vmax=1)
+pl.title('Estimated C0 matrix')
+pl.colorbar()

ot/__init__.py

@@ -43,6 +43,8 @@
                          sinkhorn_unbalanced2)
 from .da import sinkhorn_lpl1_mm
 from .sliced import sliced_wasserstein_distance, max_sliced_wasserstein_distance
+from .gromov import (gromov_wasserstein, gromov_wasserstein2,
+                        gromov_barycenters, fused_gromov_wasserstein, fused_gromov_wasserstein2)
 
 # utils functions
 from .utils import dist, unif, tic, toc, toq