1 files changed, 161 insertions, 0 deletions
diff --git a/examples/backends/plot_unmix_optim_torch.py b/examples/backends/plot_unmix_optim_torch.py
new file mode 100644
index 0000000..9ae66e9
--- /dev/null
+++ b/examples/backends/plot_unmix_optim_torch.py
@@ -0,0 +1,161 @@
+# -*- coding: utf-8 -*-
+r"""
+=================================
+Wasserstein unmixing with PyTorch
+=================================
+
+In this example we estimate mixing parameters from distributions that minimize
+the Wasserstein distance. In other words we suppose that a target
+distribution :math:`\mu^t` can be expressed as a weighted sum of source
+distributions :math:`\mu^s_k` with the following model:
+
+.. math::
+    \mu^t = \sum_{k=1}^K w_k\mu^s_k
+
+where :math:`\mathbf{w}` is a vector of size :math:`K` and belongs in the
+distribution simplex :math:`\Delta_K`.
+
+In order to estimate this weight vector we propose to optimize the Wasserstein
+distance between the model and the observed :math:`\mu^t` with respect to
+the vector. This leads to the following optimization problem:
+
+.. math::
+    \min_{\mathbf{w}\in\Delta_K} \quad W \left(\mu^t,\sum_{k=1}^K w_k\mu^s_k\right)
+
+This minimization is done in this example with a simple projected gradient
+descent in PyTorch. We use the automatic backend of POT that allows us to
+compute the Wasserstein distance with :any:`ot.emd2` with
+differentiable losses.
+
+"""
+
+# Author: Remi Flamary <remi.flamary@polytechnique.edu>
+#
+# License: MIT License
+
+# sphinx_gallery_thumbnail_number = 2
+
+import numpy as np
+import matplotlib.pylab as pl
+import ot
+import torch
+
+
+##############################################################################
+# Generate data
+# -------------
+
+#%% Data
+
+nt = 100
+nt1 = 10  #
+
+ns1 = 50
+ns = 2 * ns1
+
+rng = np.random.RandomState(2)
+
+xt = rng.randn(nt, 2) * 0.2
+xt[:nt1, 0] += 1
+xt[nt1:, 1] += 1
+
+
+xs1 = rng.randn(ns1, 2) * 0.2
+xs1[:, 0] += 1
+xs2 = rng.randn(ns1, 2) * 0.2
+xs2[:, 1] += 1
+
+xs = np.concatenate((xs1, xs2))
+
+# Sample reweighting matrix H
+H = np.zeros((ns, 2))
+H[:ns1, 0] = 1 / ns1
+H[ns1:, 1] = 1 / ns1
+# each columns sums to 1 and has weights only for samples form the
+# corresponding source distribution
+
+M = ot.dist(xs, xt)
+
+##############################################################################
+# Plot data
+# ---------
+
+#%% plot the distributions
+
+pl.figure(1)
+pl.scatter(xt[:, 0], xt[:, 1], label='Target $\mu^t$', alpha=0.5)
+pl.scatter(xs1[:, 0], xs1[:, 1], label='Source $\mu^s_1$', alpha=0.5)
+pl.scatter(xs2[:, 0], xs2[:, 1], label='Source $\mu^s_2$', alpha=0.5)
+pl.title('Sources and Target distributions')
+pl.legend()
+
+
+##############################################################################
+# Optimization of the model wrt the Wasserstein distance
+# ------------------------------------------------------
+
+
+#%% Weights optimization with gradient descent
+
+# convert numpy arrays to torch tensors
+H2 = torch.tensor(H)
+M2 = torch.tensor(M)
+
+# weights for the source distributions
+w = torch.tensor(ot.unif(2), requires_grad=True)
+
+# uniform weights for target
+b = torch.tensor(ot.unif(nt))
+
+lr = 2e-3  # learning rate
+niter = 500  # number of iterations
+losses = []  # loss along the iterations
+
+# loss for the minimal Wasserstein estimator
+
+
+def get_loss(w):
+    a = torch.mv(H2, w)  # distribution reweighting
+    return ot.emd2(a, b, M2)  # squared Wasserstein 2
+
+
+for i in range(niter):
+
+    loss = get_loss(w)
+    losses.append(float(loss))
+
+    loss.backward()
+
+    with torch.no_grad():
+        w -= lr * w.grad  # gradient step
+        w[:] = ot.utils.proj_simplex(w)  # projection on the simplex
+
+    w.grad.zero_()
+
+
+##############################################################################
+# Estimated weights and convergence of the objective
+# ---------------------------------------------------
+
+we = w.detach().numpy()
+print('Estimated mixture:', we)
+
+pl.figure(2)
+pl.semilogy(losses)
+pl.grid()
+pl.title('Wasserstein distance')
+pl.xlabel("Iterations")
+
+##############################################################################
+# Ploting the reweighted source distribution
+# ------------------------------------------
+
+pl.figure(3)
+
+# compute source weights
+ws = H.dot(we)
+
+pl.scatter(xt[:, 0], xt[:, 1], label='Target $\mu^t$', alpha=0.5)
+pl.scatter(xs[:, 0], xs[:, 1], color='C3', s=ws * 20 * ns, label='Weighted sources $\sum_{k} w_k\mu^s_k$', alpha=0.5)
+pl.title('Target and reweighted source distributions')
+pl.legend()