Merge pull request #47 from rflamary/bary

LP Wasserstein barycenter with scipy linear solver and/or cvxopt

3 files changed, 152 insertions, 1 deletions

diff --git a/ot/bregman.py b/ot/bregman.py
index 07b8660..b017c1a 100644
--- a/ot/bregman.py
+++ b/ot/bregman.py
@@ -839,11 +839,13 @@ def barycenter(A, M, reg, weights=None, numItermax=1000,
     Parameters
     ----------
     A : np.ndarray (d,n)
-        n training distributions of size d
+        n training distributions a_i of size d
     M : np.ndarray (d,d)
         loss matrix   for OT
     reg : float
         Regularization term >0
+    weights : np.ndarray (n,)
+        Weights of each histogram a_i on the simplex (barycentric coodinates)
     numItermax : int, optional
         Max number of iterations
     stopThr : float, optional
diff --git a/ot/lp/__init__.py b/ot/lp/__init__.py
index 6371feb..5dda82a 100644
--- a/ot/lp/__init__.py
+++ b/ot/lp/__init__.py
@@ -11,9 +11,12 @@ import multiprocessing
 
 import numpy as np
 
+from .import cvx
+
 # import compiled emd
 from .emd_wrap import emd_c, check_result
 from ..utils import parmap
+from .cvx import barycenter
 
 
 def emd(a, b, M, numItermax=100000, log=False):
diff --git a/ot/lp/cvx.py b/ot/lp/cvx.py
new file mode 100644
index 0000000..c8c75bc
--- /dev/null
+++ b/ot/lp/cvx.py
@@ -0,0 +1,146 @@
+# -*- coding: utf-8 -*-
+"""
+LP solvers for optimal transport using cvxopt
+"""
+
+# Author: Remi Flamary <remi.flamary@unice.fr>
+#
+# License: MIT License
+
+import numpy as np
+import scipy as sp
+import scipy.sparse as sps
+
+try:
+    import cvxopt
+    from cvxopt import solvers, matrix, spmatrix
+except ImportError:
+    cvxopt = False
+
+
+def scipy_sparse_to_spmatrix(A):
+    """Efficient conversion from scipy sparse matrix to cvxopt sparse matrix"""
+    coo = A.tocoo()
+    SP = spmatrix(coo.data.tolist(), coo.row.tolist(), coo.col.tolist(), size=A.shape)
+    return SP
+
+
+def barycenter(A, M, weights=None, verbose=False, log=False, solver='interior-point'):
+    """Compute the entropic regularized wasserstein barycenter of distributions A
+
+     The function solves the following optimization problem [16]:
+
+    .. math::
+       \mathbf{a} = arg\min_\mathbf{a} \sum_i W_{1}(\mathbf{a},\mathbf{a}_i)
+
+    where :
+
+    - :math:`W_1(\cdot,\cdot)` is the Wasserstein distance (see ot.emd.sinkhorn)
+    - :math:`\mathbf{a}_i` are training distributions in the columns of matrix :math:`\mathbf{A}`
+
+    The linear program is solved using the interior point solver from scipy.optimize.
+    If cvxopt solver if installed it can use cvxopt
+
+    Note that this problem do not scale well (both in memory and computational time).
+
+    Parameters
+    ----------
+    A : np.ndarray (d,n)
+        n training distributions a_i of size d
+    M : np.ndarray (d,d)
+        loss matrix   for OT
+    reg : float
+        Regularization term >0
+    weights : np.ndarray (n,)
+        Weights of each histogram a_i on the simplex (barycentric coodinates)
+    verbose : bool, optional
+        Print information along iterations
+    log : bool, optional
+        record log if True
+    solver : string, optional
+        the solver used, default 'interior-point' use the lp solver from
+        scipy.optimize. None, or 'glpk' or 'mosek' use the solver from cvxopt.
+
+    Returns
+    -------
+    a : (d,) ndarray
+        Wasserstein barycenter
+    log : dict
+        log dictionary return only if log==True in parameters
+
+
+    References
+    ----------
+
+    .. [16] Agueh, M., & Carlier, G. (2011). Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 43(2), 904-924.
+
+
+
+    """
+
+    if weights is None:
+        weights = np.ones(A.shape[1]) / A.shape[1]
+    else:
+        assert(len(weights) == A.shape[1])
+
+    n_distributions = A.shape[1]
+    n = A.shape[0]
+
+    n2 = n * n
+    c = np.zeros((0))
+    b_eq1 = np.zeros((0))
+    for i in range(n_distributions):
+        c = np.concatenate((c, M.ravel() * weights[i]))
+        b_eq1 = np.concatenate((b_eq1, A[:, i]))
+    c = np.concatenate((c, np.zeros(n)))
+
+    lst_idiag1 = [sps.kron(sps.eye(n), np.ones((1, n))) for i in range(n_distributions)]
+    #  row constraints
+    A_eq1 = sps.hstack((sps.block_diag(lst_idiag1), sps.coo_matrix((n_distributions * n, n))))
+
+    # columns constraints
+    lst_idiag2 = []
+    lst_eye = []
+    for i in range(n_distributions):
+        if i == 0:
+            lst_idiag2.append(sps.kron(np.ones((1, n)), sps.eye(n)))
+            lst_eye.append(-sps.eye(n))
+        else:
+            lst_idiag2.append(sps.kron(np.ones((1, n)), sps.eye(n - 1, n)))
+            lst_eye.append(-sps.eye(n - 1, n))
+
+    A_eq2 = sps.hstack((sps.block_diag(lst_idiag2), sps.vstack(lst_eye)))
+    b_eq2 = np.zeros((A_eq2.shape[0]))
+
+    # full problem
+    A_eq = sps.vstack((A_eq1, A_eq2))
+    b_eq = np.concatenate((b_eq1, b_eq2))
+
+    if not cvxopt or solver in ['interior-point']:
+        # cvxopt not installed or interior point
+
+        if solver is None:
+            solver = 'interior-point'
+
+        options = {'sparse': True, 'disp': verbose}
+        sol = sp.optimize.linprog(c, A_eq=A_eq, b_eq=b_eq, method=solver,
+                                  options=options)
+        x = sol.x
+        b = x[-n:]
+
+    else:
+
+        h = np.zeros((n_distributions * n2 + n))
+        G = -sps.eye(n_distributions * n2 + n)
+
+        sol = solvers.lp(matrix(c), scipy_sparse_to_spmatrix(G), matrix(h),
+                         A=scipy_sparse_to_spmatrix(A_eq), b=matrix(b_eq),
+                         solver=solver)
+
+        x = np.array(sol['x'])
+        b = x[-n:].ravel()
+
+    if log:
+        return b, sol
+    else:
+        return b