Free support barycenters

README.md

@@ -17,6 +17,7 @@ It provides the following solvers:
 * Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] and stabilized version [9][10] with optional GPU implementation (requires cudamat).
 * Smooth optimal transport solvers (dual and semi-dual) for KL and squared L2 regularizations [17].
 * Non regularized Wasserstein barycenters [16] with LP solver (only small scale).
+* Non regularized free support Wasserstein barycenters [20].
 * Bregman projections for Wasserstein barycenter [3] and unmixing [4].
 * Optimal transport for domain adaptation with group lasso regularization [5]
 * Conditional gradient [6] and Generalized conditional gradient for regularized OT [7].
@@ -225,3 +226,5 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016) [Stochastic Optimization for Large-scale Optimal Transport](arXiv preprint arxiv:1605.08527). Advances in Neural Information Processing Systems (2016).
 
 [19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet, A.& Blondel, M. [Large-scale Optimal Transport and Mapping Estimation](https://arxiv.org/pdf/1711.02283.pdf). International Conference on Learning Representation (2018)
+
+[20] Cuturi, M. and Doucet, A. (2014) [Fast Computation of Wasserstein Barycenters](http://proceedings.mlr.press/v32/cuturi14.html). International Conference in Machine Learning

examples/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 discributions that 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
+# -------------
+#%% parameters and data generation
+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 * 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()

ot/lp/__init__.py

@@ -17,8 +17,9 @@
 from .emd_wrap import emd_c, check_result
 from ..utils import parmap
 from .cvx import barycenter
+from ..utils import dist
 
-__all__=['emd', 'emd2', 'barycenter', 'cvx']
+__all__=['emd', 'emd2', 'barycenter', 'free_support_barycenter', 'cvx']
 
 
 def emd(a, b, M, numItermax=100000, log=False):
@@ -216,3 +217,95 @@ def f(b):
 
     res = parmap(f, [b[:, i] for i in range(nb)], processes)
     return res
+
+
+
+def free_support_barycenter(measures_locations, measures_weights, X_init, b=None, weights=None, numItermax=100, stopThr=1e-7, verbose=False, log=None):
+    """
+    Solves the free support (locations of the barycenters are optimized, not the weights) Wasserstein barycenter problem (i.e. the weighted Frechet mean for the 2-Wasserstein distance)
+
+    The function solves the Wasserstein barycenter problem when the barycenter measure is constrained to be supported on k atoms.
+    This problem is considered in [1] (Algorithm 2). There are two differences with the following codes:
+    - we do not optimize over the weights
+    - we do not do line search for the locations updates, we use i.e. theta = 1 in [1] (Algorithm 2). This can be seen as a discrete implementation of the fixed-point algorithm of [2] proposed in the continuous setting.
+
+    Parameters
+    ----------
+    measures_locations : list of (k_i,d) np.ndarray
+        The discrete support of a measure supported on k_i locations of a d-dimensional space (k_i can be different for each element of the list)
+    measures_weights : list of (k_i,) np.ndarray
+        Numpy arrays where each numpy array has k_i non-negatives values summing to one representing the weights of each discrete input measure
+
+    X_init : (k,d) np.ndarray
+        Initialization of the support locations (on k atoms) of the barycenter
+    b : (k,) np.ndarray
+        Initialization of the weights of the barycenter (non-negatives, sum to 1)
+    weights : (k,) np.ndarray
+        Initialization of the coefficients of the barycenter (non-negatives, sum to 1)
+
+    numItermax : int, optional
+        Max number of iterations
+    stopThr : float, optional
+        Stop threshol on error (>0)
+    verbose : bool, optional
+        Print information along iterations
+    log : bool, optional
+        record log if True
+
+    Returns
+    -------
+    X : (k,d) np.ndarray
+        Support locations (on k atoms) of the barycenter
+
+    References
+    ----------
+
+    .. [1] Cuturi, Marco, and Arnaud Doucet. "Fast computation of Wasserstein barycenters." International Conference on Machine Learning. 2014.
+
+    .. [2]  Álvarez-Esteban, Pedro C., et al. "A fixed-point approach to barycenters in Wasserstein space." Journal of Mathematical Analysis and Applications 441.2 (2016): 744-762.
+
+    """
+
+    iter_count = 0
+
+    N = len(measures_locations)
+    k = X_init.shape[0]
+    d = X_init.shape[1]
+    if b is None:
+        b = np.ones((k,))/k
+    if weights is None:
+        weights = np.ones((N,)) / N
+
+    X = X_init
+
+    log_dict = {}
+    displacement_square_norms = []
+
+    displacement_square_norm = stopThr + 1.
+
+    while ( displacement_square_norm > stopThr and iter_count < numItermax ):
+
+        T_sum = np.zeros((k, d))
+
+        for (measure_locations_i, measure_weights_i, weight_i) in zip(measures_locations, measures_weights, weights.tolist()):
+
+            M_i = dist(X, measure_locations_i)
+            T_i = emd(b, measure_weights_i, M_i)
+            T_sum = T_sum + weight_i * np.reshape(1. / b, (-1, 1)) * np.matmul(T_i, measure_locations_i)
+
+        displacement_square_norm = np.sum(np.square(T_sum-X))
+        if log:
+            displacement_square_norms.append(displacement_square_norm)
+
+        X = T_sum
+
+        if verbose:
+            print('iteration %d, displacement_square_norm=%f\n', iter_count, displacement_square_norm)
+
+        iter_count += 1
+
+    if log:
+        log_dict['displacement_square_norms'] = displacement_square_norms
+        return X, log_dict
+    else:
+        return X

ot/lp/cvx.py

@@ -11,6 +11,7 @@
 import scipy as sp
 import scipy.sparse as sps
 
+
 try:
     import cvxopt
     from cvxopt import solvers, matrix, spmatrix
@@ -26,7 +27,7 @@ def scipy_sparse_to_spmatrix(A):
 
 
 def barycenter(A, M, weights=None, verbose=False, log=False, solver='interior-point'):
-    """Compute the entropic regularized wasserstein barycenter of distributions A
+    """Compute the Wasserstein barycenter of distributions A
 
      The function solves the following optimization problem [16]:
 

test/test_ot.py

@@ -135,6 +135,21 @@ def test_lp_barycenter():
     np.testing.assert_allclose(bary.sum(), 1)
 
 
+def test_free_support_barycenter():
+
+    measures_locations = [np.array([-1.]).reshape((1, 1)), np.array([1.]).reshape((1, 1))]
+    measures_weights = [np.array([1.]), np.array([1.])]
+
+    X_init = np.array([-12.]).reshape((1, 1))
+
+    # obvious barycenter location between two diracs
+    bar_locations = np.array([0.]).reshape((1, 1))
+
+    X = ot.lp.free_support_barycenter(measures_locations, measures_weights, X_init)
+
+    np.testing.assert_allclose(X, bar_locations, rtol=1e-5, atol=1e-7)
+
+
 @pytest.mark.skipif(not ot.lp.cvx.cvxopt, reason="No cvxopt available")
 def test_lp_barycenter_cvxopt():