From cfdbbd21642c6082164b84db78c2ead07499a113 Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Fri, 19 Jul 2019 17:04:14 +0200 Subject: remove square in convergence check add unbalanced with stabilization add unbalanced tests with stabilization fix doctest examples add xvfb in travis remove explicit call xvfb in travis change alpha to reg_m minor flake8 remove redundant sink definitions + better doc and naming add stabilized unbalanced barycenter + add not converged warnings add test for stable barycenter add generic barycenter func + make method funcs private fix typo + add method test for barycenters fix doc examples + add xml to gitignore fix whitespace in example change logsumexp import - scipy deprecation warning fix doctest improve naming + add stable barycenter in bregman add test for stable bar + test the method arg in bregman --- .gitignore | 3 + ot/__init__.py | 18 +- ot/bregman.py | 530 +++++++++++++++++++++----------- ot/unbalanced.py | 803 +++++++++++++++++++++++++++++++++++++++--------- pytest.ini | 0 test/test_bregman.py | 72 ++++- test/test_unbalanced.py | 163 +++++++--- 7 files changed, 1205 insertions(+), 384 deletions(-) create mode 100644 pytest.ini diff --git a/.gitignore b/.gitignore index 42a9aad..dadf84c 100644 --- a/.gitignore +++ b/.gitignore @@ -59,6 +59,9 @@ coverage.xml *.mo *.pot +# xml +*.xml + # Django stuff: *.log local_settings.py diff --git a/ot/__init__.py b/ot/__init__.py index 35ae6fc..7d9615a 100644 --- a/ot/__init__.py +++ b/ot/__init__.py @@ -1,7 +1,7 @@ """ -This is the main module of the POT toolbox. It provides easy access to -a number of sub-modules and functions described below. +This is the main module of the POT toolbox. It provides easy access to +a number of sub-modules and functions described below. .. note:: @@ -14,27 +14,27 @@ a number of sub-modules and functions described below. - :any:`ot.lp` contains OT solvers for the exact (Linear Program) OT problems. - :any:`ot.smooth` contains OT solvers for the regularized (l2 and kl) smooth OT problems. - - :any:`ot.gromov` contains solvers for Gromov-Wasserstein and Fused Gromov + - :any:`ot.gromov` contains solvers for Gromov-Wasserstein and Fused Gromov Wasserstein problems. - - :any:`ot.optim` contains generic solvers OT based optimization problems + - :any:`ot.optim` contains generic solvers OT based optimization problems - :any:`ot.da` contains classes and function related to Monge mapping estimation and Domain Adaptation (DA). - :any:`ot.gpu` contains GPU (cupy) implementation of some OT solvers - - :any:`ot.dr` contains Dimension Reduction (DR) methods such as Wasserstein + - :any:`ot.dr` contains Dimension Reduction (DR) methods such as Wasserstein Discriminant Analysis. - - :any:`ot.utils` contains utility functions such as distance computation and - timing. + - :any:`ot.utils` contains utility functions such as distance computation and + timing. - :any:`ot.datasets` contains toy dataset generation functions. - :any:`ot.plot` contains visualization functions - :any:`ot.stochastic` contains stochastic solvers for regularized OT. - :any:`ot.unbalanced` contains solvers for regularized unbalanced OT. .. warning:: - The list of automatically imported sub-modules is as follows: + The list of automatically imported sub-modules is as follows: :py:mod:`ot.lp`, :py:mod:`ot.bregman`, :py:mod:`ot.optim` :py:mod:`ot.utils`, :py:mod:`ot.datasets`, :py:mod:`ot.gromov`, :py:mod:`ot.smooth` - :py:mod:`ot.stochastic` + :py:mod:`ot.stochastic` The following sub-modules are not imported due to additional dependencies: diff --git a/ot/bregman.py b/ot/bregman.py index 70e4208..2f27d58 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -7,10 +7,12 @@ Bregman projections for regularized OT # Nicolas Courty # Kilian Fatras # Titouan Vayer +# Hicham Janati # # License: MIT License import numpy as np +import warnings from .utils import unif, dist @@ -31,7 +33,7 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, \gamma\geq 0 where : - - M is the (ns,nt) metric cost matrix + - M is the (dim_a, n_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -40,12 +42,12 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, Parameters ---------- - a : ndarray, shape (ns,) + a : ndarray, shape (dim_a,) samples weights in the source domain - b : ndarray, shape (nt,) or ndarray, shape (nt, nbb) + b : ndarray, shape (dim_b,) or ndarray, shape (dim_b, n_hists) samples in the target domain, compute sinkhorn with multiple targets and fixed M if b is a matrix (return OT loss + dual variables in log) - M : ndarray, shape (ns, nt) + M : ndarray, shape (dim_a, n_b) loss matrix reg : float Regularization term >0 @@ -64,7 +66,7 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, Returns ------- - gamma : ndarray, shape (ns, nt) + gamma : ndarray, shape (dim_a, n_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -103,30 +105,23 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, """ if method.lower() == 'sinkhorn': - def sink(): - return sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, **kwargs) + return _sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log, + **kwargs) elif method.lower() == 'greenkhorn': - def sink(): - return greenkhorn(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log) + return _greenkhorn(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log) elif method.lower() == 'sinkhorn_stabilized': - def sink(): - return sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, **kwargs) + return _sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) elif method.lower() == 'sinkhorn_epsilon_scaling': - def sink(): - return sinkhorn_epsilon_scaling( - a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, **kwargs) + return _sinkhorn_epsilon_scaling(a, b, M, reg, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) else: - print('Warning : unknown method using classic Sinkhorn Knopp') - - def sink(): - return sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, **kwargs) - - return sink() + raise ValueError("Unknown method '%s'." % method) def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, @@ -146,7 +141,7 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, \gamma\geq 0 where : - - M is the (ns,nt) metric cost matrix + - M is the (dim_a, n_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -155,12 +150,12 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, Parameters ---------- - a : ndarray, shape (ns,) + a : ndarray, shape (dim_a,) samples weights in the source domain - b : ndarray, shape (nt,) or ndarray, shape (nt, nbb) + b : ndarray, shape (dim_b,) or ndarray, shape (dim_b, n_hists) samples in the target domain, compute sinkhorn with multiple targets and fixed M if b is a matrix (return OT loss + dual variables in log) - M : ndarray, shape (ns, nt) + M : ndarray, shape (dim_a, n_b) loss matrix reg : float Regularization term >0 @@ -218,35 +213,25 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, ot.bregman.sinkhorn_epsilon_scaling: Sinkhorn with epslilon scaling [9][10] """ - + b = np.asarray(b, dtype=np.float64) + if len(b.shape) < 2: + b = b[:, None] if method.lower() == 'sinkhorn': - def sink(): - return sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, **kwargs) + return _sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log, **kwargs) elif method.lower() == 'sinkhorn_stabilized': - def sink(): - return sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, **kwargs) + return _sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log, **kwargs) elif method.lower() == 'sinkhorn_epsilon_scaling': - def sink(): - return sinkhorn_epsilon_scaling( - a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, **kwargs) + return _sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) else: - print('Warning : unknown method using classic Sinkhorn Knopp') - - def sink(): - return sinkhorn_knopp(a, b, M, reg, **kwargs) + raise ValueError("Unknown method '%s'." % method) - b = np.asarray(b, dtype=np.float64) - if len(b.shape) < 2: - b = b[:, None] - return sink() - - -def sinkhorn_knopp(a, b, M, reg, numItermax=1000, - stopThr=1e-9, verbose=False, log=False, **kwargs): +def _sinkhorn_knopp(a, b, M, reg, numItermax=1000, + stopThr=1e-9, verbose=False, log=False, **kwargs): r""" Solve the entropic regularization optimal transport problem and return the OT matrix @@ -262,7 +247,7 @@ def sinkhorn_knopp(a, b, M, reg, numItermax=1000, \gamma\geq 0 where : - - M is the (ns,nt) metric cost matrix + - M is the (dim_a, n_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -271,12 +256,12 @@ def sinkhorn_knopp(a, b, M, reg, numItermax=1000, Parameters ---------- - a : ndarray, shape (ns,) + a : ndarray, shape (dim_a,) samples weights in the source domain - b : ndarray, shape (nt,) or ndarray, shape (nt, nbb) + b : ndarray, shape (dim_b,) or ndarray, shape (dim_b, n_hists) samples in the target domain, compute sinkhorn with multiple targets and fixed M if b is a matrix (return OT loss + dual variables in log) - M : ndarray, shape (ns, nt) + M : ndarray, shape (dim_a, n_b) loss matrix reg : float Regularization term >0 @@ -291,7 +276,7 @@ def sinkhorn_knopp(a, b, M, reg, numItermax=1000, Returns ------- - gamma : ndarray, shape (ns, nt) + gamma : ndarray, shape (dim_a, n_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -331,25 +316,25 @@ def sinkhorn_knopp(a, b, M, reg, numItermax=1000, b = np.ones((M.shape[1],), dtype=np.float64) / M.shape[1] # init data - Nini = len(a) - Nfin = len(b) + dim_a = len(a) + dim_b = len(b) if len(b.shape) > 1: - nbb = b.shape[1] + n_hists = b.shape[1] else: - nbb = 0 + n_hists = 0 if log: log = {'err': []} # we assume that no distances are null except those of the diagonal of # distances - if nbb: - u = np.ones((Nini, nbb)) / Nini - v = np.ones((Nfin, nbb)) / Nfin + if n_hists: + u = np.ones((dim_a, n_hists)) / dim_a + v = np.ones((dim_b, n_hists)) / dim_b else: - u = np.ones(Nini) / Nini - v = np.ones(Nfin) / Nfin + u = np.ones(dim_a) / dim_a + v = np.ones(dim_b) / dim_b # print(reg) @@ -384,13 +369,12 @@ def sinkhorn_knopp(a, b, M, reg, numItermax=1000, if cpt % 10 == 0: # we can speed up the process by checking for the error only all # the 10th iterations - if nbb: - err = np.sum((u - uprev)**2) / np.sum((u)**2) + \ - np.sum((v - vprev)**2) / np.sum((v)**2) + if n_hists: + np.einsum('ik,ij,jk->jk', u, K, v, out=tmp2) else: # compute right marginal tmp2= (diag(u)Kdiag(v))^T1 np.einsum('i,ij,j->j', u, K, v, out=tmp2) - err = np.linalg.norm(tmp2 - b)**2 # violation of marginal + err = np.linalg.norm(tmp2 - b) # violation of marginal if log: log['err'].append(err) @@ -404,7 +388,7 @@ def sinkhorn_knopp(a, b, M, reg, numItermax=1000, log['u'] = u log['v'] = v - if nbb: # return only loss + if n_hists: # return only loss res = np.einsum('ik,ij,jk,ij->k', u, K, v, M) if log: return res, log @@ -419,7 +403,7 @@ def sinkhorn_knopp(a, b, M, reg, numItermax=1000, return u.reshape((-1, 1)) * K * v.reshape((1, -1)) -def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log=False): +def _greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log=False): r""" Solve the entropic regularization optimal transport problem and return the OT matrix @@ -443,7 +427,7 @@ def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log= \gamma\geq 0 where : - - M is the (ns,nt) metric cost matrix + - M is the (dim_a, n_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -451,12 +435,12 @@ def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log= Parameters ---------- - a : ndarray, shape (ns,) + a : ndarray, shape (dim_a,) samples weights in the source domain - b : ndarray, shape (nt,) or ndarray, shape (nt, nbb) + b : ndarray, shape (dim_b,) or ndarray, shape (dim_b, n_hists) samples in the target domain, compute sinkhorn with multiple targets and fixed M if b is a matrix (return OT loss + dual variables in log) - M : ndarray, shape (ns, nt) + M : ndarray, shape (dim_a, n_b) loss matrix reg : float Regularization term >0 @@ -469,7 +453,7 @@ def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log= Returns ------- - gamma : ndarray, shape (ns, nt) + gamma : ndarray, shape (dim_a, n_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -481,7 +465,7 @@ def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log= >>> a=[.5, .5] >>> b=[.5, .5] >>> M=[[0., 1.], [1., 0.]] - >>> ot.bregman.greenkhorn(a, b, M, 1) + >>> ot.bregman._greenkhorn(a, b, M, 1) array([[0.36552929, 0.13447071], [0.13447071, 0.36552929]]) @@ -509,16 +493,16 @@ def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log= if len(b) == 0: b = np.ones((M.shape[1],), dtype=np.float64) / M.shape[1] - n = a.shape[0] - m = b.shape[0] + dim_a = a.shape[0] + dim_b = b.shape[0] # Next 3 lines equivalent to K= np.exp(-M/reg), but faster to compute K = np.empty_like(M) np.divide(M, -reg, out=K) np.exp(K, out=K) - u = np.full(n, 1. / n) - v = np.full(m, 1. / m) + u = np.full(dim_a, 1. / dim_a) + v = np.full(dim_b, 1. / dim_b) G = u[:, np.newaxis] * K * v[np.newaxis, :] viol = G.sum(1) - a @@ -571,8 +555,9 @@ def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log= return G -def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, - warmstart=None, verbose=False, print_period=20, log=False, **kwargs): +def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, + warmstart=None, verbose=False, print_period=20, + log=False, **kwargs): r""" Solve the entropic regularization OT problem with log stabilization @@ -588,7 +573,7 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, \gamma\geq 0 where : - - M is the (ns,nt) metric cost matrix + - M is the (dim_a, n_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -599,11 +584,11 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, Parameters ---------- - a : ndarray, shape (ns,) + a : ndarray, shape (dim_a,) samples weights in the source domain - b : ndarray, shape (nt,) + b : ndarray, shape (dim_b,) samples in the target domain - M : ndarray, shape (ns, nt) + M : ndarray, shape (dim_a, n_b) loss matrix reg : float Regularization term >0 @@ -622,7 +607,7 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, Returns ------- - gamma : ndarray, shape (ns, nt) + gamma : ndarray, shape (dim_a, n_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -634,7 +619,7 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, >>> a=[.5,.5] >>> b=[.5,.5] >>> M=[[0.,1.],[1.,0.]] - >>> ot.bregman.sinkhorn_stabilized(a,b,M,1) + >>> ot.bregman._sinkhorn_stabilized(a, b, M, 1) array([[0.36552929, 0.13447071], [0.13447071, 0.36552929]]) @@ -667,10 +652,10 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, # test if multiple target if len(b.shape) > 1: - nbb = b.shape[1] + n_hists = b.shape[1] a = a[:, np.newaxis] else: - nbb = 0 + n_hists = 0 # init data na = len(a) @@ -687,8 +672,8 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, else: alpha, beta = warmstart - if nbb: - u, v = np.ones((na, nbb)) / na, np.ones((nb, nbb)) / nb + if n_hists: + u, v = np.ones((na, n_hists)) / na, np.ones((nb, n_hists)) / nb else: u, v = np.ones(na) / na, np.ones(nb) / nb @@ -720,13 +705,13 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, # remove numerical problems and store them in K if np.abs(u).max() > tau or np.abs(v).max() > tau: - if nbb: + if n_hists: alpha, beta = alpha + reg * \ np.max(np.log(u), 1), beta + reg * np.max(np.log(v)) else: alpha, beta = alpha + reg * np.log(u), beta + reg * np.log(v) - if nbb: - u, v = np.ones((na, nbb)) / na, np.ones((nb, nbb)) / nb + if n_hists: + u, v = np.ones((na, n_hists)) / na, np.ones((nb, n_hists)) / nb else: u, v = np.ones(na) / na, np.ones(nb) / nb K = get_K(alpha, beta) @@ -734,12 +719,15 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, if cpt % print_period == 0: # we can speed up the process by checking for the error only all # the 10th iterations - if nbb: - err = np.sum((u - uprev)**2) / np.sum((u)**2) + \ - np.sum((v - vprev)**2) / np.sum((v)**2) + if n_hists: + err_u = abs(u - uprev).max() + err_u /= max(abs(u).max(), abs(uprev).max(), 1.) + err_v = abs(v - vprev).max() + err_v /= max(abs(v).max(), abs(vprev).max(), 1.) + err = 0.5 * (err_u + err_v) else: transp = get_Gamma(alpha, beta, u, v) - err = np.linalg.norm((np.sum(transp, axis=0) - b))**2 + err = np.linalg.norm((np.sum(transp, axis=0) - b)) if log: log['err'].append(err) @@ -766,7 +754,7 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, cpt = cpt + 1 if log: - if nbb: + if n_hists: alpha = alpha[:, None] beta = beta[:, None] logu = alpha / reg + np.log(u) @@ -776,26 +764,28 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, log['alpha'] = alpha + reg * np.log(u) log['beta'] = beta + reg * np.log(v) log['warmstart'] = (log['alpha'], log['beta']) - if nbb: - res = np.zeros((nbb)) - for i in range(nbb): + if n_hists: + res = np.zeros((n_hists)) + for i in range(n_hists): res[i] = np.sum(get_Gamma(alpha, beta, u[:, i], v[:, i]) * M) return res, log else: return get_Gamma(alpha, beta, u, v), log else: - if nbb: - res = np.zeros((nbb)) - for i in range(nbb): + if n_hists: + res = np.zeros((n_hists)) + for i in range(n_hists): res[i] = np.sum(get_Gamma(alpha, beta, u[:, i], v[:, i]) * M) return res else: return get_Gamma(alpha, beta, u, v) -def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, numInnerItermax=100, - tau=1e3, stopThr=1e-9, warmstart=None, verbose=False, print_period=10, log=False, **kwargs): +def _sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, + numInnerItermax=100, tau=1e3, stopThr=1e-9, + warmstart=None, verbose=False, print_period=10, + log=False, **kwargs): r""" Solve the entropic regularization optimal transport problem with log stabilization and epsilon scaling. @@ -812,7 +802,7 @@ def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, numInne \gamma\geq 0 where : - - M is the (ns,nt) metric cost matrix + - M is the (dim_a, n_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -823,18 +813,16 @@ def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, numInne Parameters ---------- - a : ndarray, shape (ns,) + a : ndarray, shape (dim_a,) samples weights in the source domain - b : ndarray, shape (nt,) + b : ndarray, shape (dim_b,) samples in the target domain - M : ndarray, shape (ns, nt) + M : ndarray, shape (dim_a, n_b) loss matrix reg : float Regularization term >0 tau : float thershold for max value in u or v for log scaling - tau : float - thershold for max value in u or v for log scaling warmstart : tuple of vectors if given then sarting values for alpha an beta log scalings numItermax : int, optional @@ -852,7 +840,7 @@ def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, numInne Returns ------- - gamma : ndarray, shape (ns, nt) + gamma : ndarray, shape (dim_a, n_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -864,7 +852,7 @@ def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, numInne >>> a=[.5, .5] >>> b=[.5, .5] >>> M=[[0., 1.], [1., 0.]] - >>> ot.bregman.sinkhorn_epsilon_scaling(a, b, M, 1) + >>> ot.bregman._sinkhorn_epsilon_scaling(a, b, M, 1) array([[0.36552929, 0.13447071], [0.13447071, 0.36552929]]) @@ -893,8 +881,8 @@ def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, numInne b = np.ones((M.shape[1],), dtype=np.float64) / M.shape[1] # init data - na = len(a) - nb = len(b) + dim_a = len(a) + dim_b = len(b) # nrelative umerical precision with 64 bits numItermin = 35 @@ -907,14 +895,14 @@ def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, numInne # we assume that no distances are null except those of the diagonal of # distances if warmstart is None: - alpha, beta = np.zeros(na), np.zeros(nb) + alpha, beta = np.zeros(dim_a), np.zeros(dim_b) else: alpha, beta = warmstart def get_K(alpha, beta): """log space computation""" - return np.exp(-(M - alpha.reshape((na, 1)) - - beta.reshape((1, nb))) / reg) + return np.exp(-(M - alpha.reshape((dim_a, 1)) + - beta.reshape((1, dim_b))) / reg) # print(np.min(K)) def get_reg(n): # exponential decreasing @@ -927,7 +915,7 @@ def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, numInne regi = get_reg(cpt) - G, logi = sinkhorn_stabilized(a, b, M, regi, numItermax=numInnerItermax, stopThr=1e-9, warmstart=( + G, logi = _sinkhorn_stabilized(a, b, M, regi, numItermax=numInnerItermax, stopThr=1e-9, warmstart=( alpha, beta), verbose=False, print_period=20, tau=tau, log=True) alpha = logi['alpha'] @@ -986,8 +974,8 @@ def projC(gamma, q): return np.multiply(gamma, q / np.maximum(np.sum(gamma, axis=0), 1e-10)) -def barycenter(A, M, reg, weights=None, numItermax=1000, - stopThr=1e-4, verbose=False, log=False): +def barycenter(A, M, reg, weights=None, method="sinkhorn", numItermax=10000, + stopThr=1e-4, verbose=False, log=False, **kwargs): r"""Compute the entropic regularized wasserstein barycenter of distributions A The function solves the following optimization problem: @@ -1005,13 +993,15 @@ def barycenter(A, M, reg, weights=None, numItermax=1000, Parameters ---------- - A : ndarray, shape (d,n) - n training distributions a_i of size d - M : ndarray, shape (d,d) - loss matrix for OT + A : ndarray, shape (dim, n_hists) + n_hists training distributions a_i of size dim + M : ndarray, shape (dim, dim) + loss matrix for OT reg : float - Regularization term >0 - weights : ndarray, shape (n,) + Regularization term > 0 + method : str (optional) + method used for the solver either 'sinkhorn' or 'sinkhorn_stabilized' + weights : ndarray, shape (n_hists,) Weights of each histogram a_i on the simplex (barycentric coodinates) numItermax : int, optional Max number of iterations @@ -1025,7 +1015,7 @@ def barycenter(A, M, reg, weights=None, numItermax=1000, Returns ------- - a : (d,) ndarray + a : (dim,) ndarray Wasserstein barycenter log : dict log dictionary return only if log==True in parameters @@ -1036,8 +1026,70 @@ def barycenter(A, M, reg, weights=None, numItermax=1000, .. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015). Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138. + """ + + if method.lower() == 'sinkhorn': + return _barycenter(A, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log, + **kwargs) + elif method.lower() == 'sinkhorn_stabilized': + return _barycenter_stabilized(A, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) + else: + raise ValueError("Unknown method '%s'." % method) + + +def _barycenter(A, M, reg, weights=None, numItermax=1000, + stopThr=1e-4, verbose=False, log=False): + r"""Compute the entropic regularized wasserstein barycenter of distributions A + + The function solves the following optimization problem: + + .. math:: + \mathbf{a} = arg\min_\mathbf{a} \sum_i W_{reg}(\mathbf{a},\mathbf{a}_i) + + where : + + - :math:`W_{reg}(\cdot,\cdot)` is the entropic regularized Wasserstein distance (see ot.bregman.sinkhorn) + - :math:`\mathbf{a}_i` are training distributions in the columns of matrix :math:`\mathbf{A}` + - reg and :math:`\mathbf{M}` are respectively the regularization term and the cost matrix for OT + + The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [3]_ + + Parameters + ---------- + A : ndarray, shape (dim, n_hists) + n_hists training distributions a_i of size dim + M : ndarray, shape (dim, dim) + loss matrix for OT + reg : float + Regularization term > 0 + weights : ndarray, shape (n_hists,) + Weights of each histogram a_i on the simplex (barycentric coodinates) + 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 + ------- + a : (dim,) ndarray + Wasserstein barycenter + log : dict + log dictionary return only if log==True in parameters + + + References + ---------- + + .. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015). Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138. + """ if weights is None: @@ -1082,6 +1134,136 @@ def barycenter(A, M, reg, weights=None, numItermax=1000, return geometricBar(weights, UKv) +def _barycenter_stabilized(A, M, reg, tau=1e10, weights=None, numItermax=1000, + stopThr=1e-4, verbose=False, log=False): + r"""Compute the entropic regularized wasserstein barycenter of distributions A + with stabilization. + + The function solves the following optimization problem: + + .. math:: + \mathbf{a} = arg\min_\mathbf{a} \sum_i W_{reg}(\mathbf{a},\mathbf{a}_i) + + where : + + - :math:`W_{reg}(\cdot,\cdot)` is the entropic regularized Wasserstein distance (see ot.bregman.sinkhorn) + - :math:`\mathbf{a}_i` are training distributions in the columns of matrix :math:`\mathbf{A}` + - reg and :math:`\mathbf{M}` are respectively the regularization term and the cost matrix for OT + + The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [3]_ + + Parameters + ---------- + A : ndarray, shape (dim, n_hists) + n_hists training distributions a_i of size dim + M : ndarray, shape (dim, dim) + loss matrix for OT + reg : float + Regularization term > 0 + tau : float + thershold for max value in u or v for log scaling + weights : ndarray, shape (n_hists,) + Weights of each histogram a_i on the simplex (barycentric coodinates) + 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 + ------- + a : (dim,) ndarray + Wasserstein barycenter + log : dict + log dictionary return only if log==True in parameters + + + References + ---------- + + .. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015). Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138. + + """ + + dim, n_hists = A.shape + if weights is None: + weights = np.ones(n_hists) / n_hists + else: + assert(len(weights) == A.shape[1]) + + if log: + log = {'err': []} + + u = np.ones((dim, n_hists)) / dim + v = np.ones((dim, n_hists)) / dim + + # print(reg) + # Next 3 lines equivalent to K= np.exp(-M/reg), but faster to compute + K = np.empty(M.shape, dtype=M.dtype) + np.divide(M, -reg, out=K) + np.exp(K, out=K) + + cpt = 0 + err = 1. + alpha = np.zeros(dim) + beta = np.zeros(dim) + q = np.ones(dim) / dim + while (err > stopThr and cpt < numItermax): + qprev = q + Kv = K.dot(v) + u = A / (Kv + 1e-16) + Ktu = K.T.dot(u) + q = geometricBar(weights, Ktu) + Q = q[:, None] + v = Q / (Ktu + 1e-16) + absorbing = False + if (u > tau).any() or (v > tau).any(): + absorbing = True + print("YEAH absorbing") + alpha = alpha + reg * np.log(np.max(u, 1)) + beta = beta + reg * np.log(np.max(v, 1)) + K = np.exp((alpha[:, None] + beta[None, :] - + M) / reg) + v = np.ones_like(v) + Kv = K.dot(v) + if (np.any(Ktu == 0.) + or np.any(np.isnan(u)) or np.any(np.isnan(v)) + or np.any(np.isinf(u)) or np.any(np.isinf(v))): + # we have reached the machine precision + # come back to previous solution and quit loop + warnings.warn('Numerical errors at iteration %s' % cpt) + q = qprev + break + if (cpt % 10 == 0 and not absorbing) or cpt == 0: + # we can speed up the process by checking for the error only all + # the 10th iterations + err = abs(u * Kv - A).max() + if log: + log['err'].append(err) + if verbose: + if cpt % 50 == 0: + print( + '{:5s}|{:12s}'.format('It.', 'Err') + '\n' + '-' * 19) + print('{:5d}|{:8e}|'.format(cpt, err)) + + cpt += 1 + if err > stopThr: + warnings.warn("Stabilized Unbalanced Sinkhorn did not converge." + + "Try a larger entropy `reg`" + + "Or a larger absorption threshold `tau`.") + if log: + log['niter'] = cpt + log['logu'] = np.log(u + 1e-16) + log['logv'] = np.log(v + 1e-16) + return q, log + else: + return q + + def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1e-9, stabThr=1e-30, verbose=False, log=False): r"""Compute the entropic regularized wasserstein barycenter of distributions A where A is a collection of 2D images. @@ -1101,16 +1283,16 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1 Parameters ---------- - A : ndarray, shape (n, w, h) - n distributions (2D images) of size w x h + A : ndarray, shape (n_hists, width, height) + n distributions (2D images) of size width x height reg : float Regularization term >0 - weights : ndarray, shape (n,) + weights : ndarray, shape (n_hists,) Weights of each image on the simplex (barycentric coodinates) numItermax : int, optional Max number of iterations stopThr : float, optional - Stop threshol on error (>0) + Stop threshol on error (> 0) stabThr : float, optional Stabilization threshold to avoid numerical precision issue verbose : bool, optional @@ -1120,7 +1302,7 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1 Returns ------- - a : ndarray, shape (w, h) + a : ndarray, shape (width, height) 2D Wasserstein barycenter log : dict log dictionary return only if log==True in parameters @@ -1214,15 +1396,15 @@ def unmix(a, D, M, M0, h0, reg, reg0, alpha, numItermax=1000, Parameters ---------- - a : ndarray, shape (d) + a : ndarray, shape (n_observed) observed distribution - D : ndarray, shape (d, n) + D : ndarray, shape (dim, dim) dictionary matrix - M : ndarray, shape (d, d) + M : ndarray, shape (dim, dim) loss matrix - M0 : ndarray, shape (n, n) + M0 : ndarray, shape (n_observed, n_observed) loss matrix - h0 : ndarray, shape (n,) + h0 : ndarray, shape (dim,) prior on h reg : float Regularization term >0 (Wasserstein data fitting) @@ -1242,7 +1424,7 @@ def unmix(a, D, M, M0, h0, reg, reg0, alpha, numItermax=1000, Returns ------- - a : ndarray, shape (d,) + a : ndarray, shape (dim,) Wasserstein barycenter log : dict log dictionary return only if log==True in parameters @@ -1315,22 +1497,22 @@ def empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numI \gamma\geq 0 where : - - :math:`M` is the (ns,nt) metric cost matrix + - :math:`M` is the (dim_a, n_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - :math:`a` and :math:`b` are source and target weights (sum to 1) Parameters ---------- - X_s : ndarray, shape (ns, d) + X_s : ndarray, shape (dim_a, d) samples in the source domain - X_t : ndarray, shape (nt, d) + X_t : ndarray, shape (dim_b, d) samples in the target domain reg : float Regularization term >0 - a : ndarray, shape (ns,) + a : ndarray, shape (dim_a,) samples weights in the source domain - b : ndarray, shape (nt,) + b : ndarray, shape (dim_b,) samples weights in the target domain numItermax : int, optional Max number of iterations @@ -1344,7 +1526,7 @@ def empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numI Returns ------- - gamma : ndarray, shape (ns, nt) + gamma : ndarray, shape (dim_a, n_b) Regularized optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -1352,11 +1534,11 @@ def empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numI Examples -------- - >>> n_s = 2 - >>> n_t = 2 + >>> n_a = 2 + >>> n_b = 2 >>> reg = 0.1 - >>> X_s = np.reshape(np.arange(n_s), (n_s, 1)) - >>> X_t = np.reshape(np.arange(0, n_t), (n_t, 1)) + >>> X_s = np.reshape(np.arange(n_a), (dim_a, 1)) + >>> X_t = np.reshape(np.arange(0, n_b), (dim_b, 1)) >>> empirical_sinkhorn(X_s, X_t, reg, verbose=False) # doctest: +NORMALIZE_WHITESPACE array([[4.99977301e-01, 2.26989344e-05], [2.26989344e-05, 4.99977301e-01]]) @@ -1405,22 +1587,22 @@ def empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', num \gamma\geq 0 where : - - :math:`M` is the (ns,nt) metric cost matrix + - :math:`M` is the (dim_a, n_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - :math:`a` and :math:`b` are source and target weights (sum to 1) Parameters ---------- - X_s : ndarray, shape (ns, d) + X_s : ndarray, shape (n_samples_a, dim) samples in the source domain - X_t : ndarray, shape (nt, d) + X_t : ndarray, shape (n_samples_b, d) samples in the target domain reg : float Regularization term >0 - a : ndarray, shape (ns,) + a : ndarray, shape (n_samples_a,) samples weights in the source domain - b : ndarray, shape (nt,) + b : ndarray, shape (n_samples_b,) samples weights in the target domain numItermax : int, optional Max number of iterations @@ -1434,7 +1616,7 @@ def empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', num Returns ------- - gamma : ndarray, shape (ns, nt) + gamma : ndarray, shape (n_samples_a, n_samples_b) Regularized optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -1442,11 +1624,11 @@ def empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', num Examples -------- - >>> n_s = 2 - >>> n_t = 2 + >>> n_a = 2 + >>> n_b = 2 >>> reg = 0.1 - >>> X_s = np.reshape(np.arange(n_s), (n_s, 1)) - >>> X_t = np.reshape(np.arange(0, n_t), (n_t, 1)) + >>> X_s = np.reshape(np.arange(n_samples_a), (n_samples_a, 1)) + >>> X_t = np.reshape(np.arange(0, n_samples_b), (n_samples_b, 1)) >>> empirical_sinkhorn2(X_s, X_t, reg, verbose=False) array([4.53978687e-05]) @@ -1513,22 +1695,22 @@ def empirical_sinkhorn_divergence(X_s, X_t, reg, a=None, b=None, metric='sqeucli \gamma_b\geq 0 where : - - :math:`M` (resp. :math:`M_a, M_b`) is the (ns,nt) metric cost matrix (resp (ns, ns) and (nt, nt)) + - :math:`M` (resp. :math:`M_a, M_b`) is the (dim_a, n_b) metric cost matrix (resp (dim_a, ns) and (dim_b, nt)) - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - :math:`a` and :math:`b` are source and target weights (sum to 1) Parameters ---------- - X_s : ndarray, shape (ns, d) + X_s : ndarray, shape (n_samples_a, dim) samples in the source domain - X_t : ndarray, shape (nt, d) + X_t : ndarray, shape (n_samples_b, dim) samples in the target domain reg : float Regularization term >0 - a : ndarray, shape (ns,) + a : ndarray, shape (n_samples_a,) samples weights in the source domain - b : ndarray, shape (nt,) + b : ndarray, shape (n_samples_b,) samples weights in the target domain numItermax : int, optional Max number of iterations @@ -1541,18 +1723,18 @@ def empirical_sinkhorn_divergence(X_s, X_t, reg, a=None, b=None, metric='sqeucli Returns ------- - gamma : ndarray, shape (ns, nt) + gamma : ndarray, shape (n_samples_a, n_samples_b) Regularized optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters Examples -------- - >>> n_s = 2 - >>> n_t = 4 + >>> n_a = 2 + >>> n_b = 4 >>> reg = 0.1 - >>> X_s = np.reshape(np.arange(n_s), (n_s, 1)) - >>> X_t = np.reshape(np.arange(0, n_t), (n_t, 1)) + >>> X_s = np.reshape(np.arange(n_samples_a), (n_samples_a, 1)) + >>> X_t = np.reshape(np.arange(0, n_samples_b), (n_samples_b, 1)) >>> empirical_sinkhorn_divergence(X_s, X_t, reg) # doctest: +ELLIPSIS array([1.499...]) diff --git a/ot/unbalanced.py b/ot/unbalanced.py index 0f0692e..3f71d28 100644 --- a/ot/unbalanced.py +++ b/ot/unbalanced.py @@ -9,51 +9,56 @@ Regularized Unbalanced OT from __future__ import division import warnings import numpy as np +from scipy.special import logsumexp + # from .utils import unif, dist -def sinkhorn_unbalanced(a, b, M, reg, alpha, method='sinkhorn', numItermax=1000, - stopThr=1e-9, verbose=False, log=False, **kwargs): +def sinkhorn_unbalanced(a, b, M, reg, reg_m, method='sinkhorn', numItermax=1000, + stopThr=1e-6, verbose=False, log=False, **kwargs): r""" - Solve the unbalanced entropic regularization optimal transport problem and return the loss + Solve the unbalanced entropic regularization optimal transport problem + and return the OT plan The function solves the following optimization problem: .. math:: - W = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma) + \\alpha KL(\gamma 1, a) + \\alpha KL(\gamma^T 1, b) + W = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma) + reg_m KL(\gamma 1, a) + reg_m KL(\gamma^T 1, b) s.t. \gamma\geq 0 where : - - M is the (ns, nt) metric cost matrix - - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - a and b are source and target weights + - M is the (dim_a, dim_b) metric cost matrix + - :math:`\Omega` is the entropic regularization + term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` + - a and b are source and target unbalanced distributions - KL is the Kullback-Leibler divergence - The algorithm used for solving the problem is the generalized Sinkhorn-Knopp matrix scaling algorithm as proposed in [10, 23]_ + The algorithm used for solving the problem is the generalized + Sinkhorn-Knopp matrix scaling algorithm as proposed in [10, 23]_ Parameters ---------- - a : np.ndarray (ns,) - samples weights in the source domain - b : np.ndarray (nt,) or np.ndarray (nt,n_hists) - samples in the target domain, compute sinkhorn with multiple targets - and fixed M if b is a matrix (return OT loss + dual variables in log) - M : np.ndarray (ns, nt) + a : np.ndarray (dim_a,) + Unnormalized histogram of dimension dim_a + b : np.ndarray (dim_b,) or np.ndarray (dim_b, n_hists) + One or multiple unnormalized histograms of dimension dim_b + If many, compute all the OT distances (a, b_i) + M : np.ndarray (dim_a, dim_b) loss matrix reg : float Entropy regularization term > 0 - alpha : float + reg_m: float Marginal relaxation term > 0 method : str method used for the solver either 'sinkhorn', 'sinkhorn_stabilized' or - 'sinkhorn_epsilon_scaling', see those function for specific parameters + 'sinkhorn_reg_scaling', see those function for specific parameters numItermax : int, optional Max number of iterations stopThr : float, optional - Stop threshol on error (> 0) + Stop threshol on error (>0) verbose : bool, optional Print information along iterations log : bool, optional @@ -62,10 +67,16 @@ def sinkhorn_unbalanced(a, b, M, reg, alpha, method='sinkhorn', numItermax=1000, Returns ------- - W : (nt) ndarray or float - Optimal transportation matrix for the given parameters - log : dict - log dictionary return only if log==True in parameters + if n_hists == 1: + gamma : (dim_a x dim_b) ndarray + Optimal transportation matrix for the given parameters + log : dict + log dictionary returned only if `log` is `True` + else: + ot_distance : (n_hists,) ndarray + the OT distance between `a` and each of the histograms `b_i` + log : dict + log dictionary returned only if `log` is `True` Examples -------- @@ -82,83 +93,96 @@ def sinkhorn_unbalanced(a, b, M, reg, alpha, method='sinkhorn', numItermax=1000, References ---------- - .. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013 + .. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal + Transport, Advances in Neural Information Processing Systems + (NIPS) 26, 2013 - .. [9] Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519. + .. [9] Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms for + Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519. - .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816. + .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). + Scaling algorithms for unbalanced transport problems. arXiv preprint + arXiv:1607.05816. - .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss, Advances in Neural Information Processing Systems (NIPS) 2015 + .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : + Learning with a Wasserstein Loss, Advances in Neural Information + Processing Systems (NIPS) 2015 See Also -------- ot.unbalanced.sinkhorn_knopp_unbalanced : Unbalanced Classic Sinkhorn [10] - ot.unbalanced.sinkhorn_stabilized_unbalanced: Unbalanced Stabilized sinkhorn [9][10] - ot.unbalanced.sinkhorn_epsilon_scaling_unbalanced: Unbalanced Sinkhorn with epslilon scaling [9][10] + ot.unbalanced.sinkhorn_stabilized_unbalanced: + Unbalanced Stabilized sinkhorn [9][10] + ot.unbalanced.sinkhorn_reg_scaling_unbalanced: + Unbalanced Sinkhorn with epslilon scaling [9][10] """ if method.lower() == 'sinkhorn': - def sink(): - return sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) - - elif method.lower() in ['sinkhorn_stabilized', 'sinkhorn_epsilon_scaling']: + return _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) + + elif method.lower() == 'sinkhorn_stabilized': + return _sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, + verbose=verbose, + log=log, **kwargs) + elif method.lower() in ['sinkhorn_reg_scaling']: warnings.warn('Method not implemented yet. Using classic Sinkhorn Knopp') - - def sink(): - return sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) else: - raise ValueError('Unknown method. Using classic Sinkhorn Knopp') - - return sink() + raise ValueError("Unknown method '%s'." % method) -def sinkhorn_unbalanced2(a, b, M, reg, alpha, method='sinkhorn', - numItermax=1000, stopThr=1e-9, verbose=False, +def sinkhorn_unbalanced2(a, b, M, reg, reg_m, method='sinkhorn', + numItermax=1000, stopThr=1e-6, verbose=False, log=False, **kwargs): r""" - Solve the entropic regularization unbalanced optimal transport problem and return the loss + Solve the entropic regularization unbalanced optimal transport problem and + return the loss The function solves the following optimization problem: .. math:: - W = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma) + \\alpha KL(\gamma 1, a) + \\alpha KL(\gamma^T 1, b) + W = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma) + reg_m KL(\gamma 1, a) + reg_m KL(\gamma^T 1, b) s.t. \gamma\geq 0 where : - - M is the (ns, nt) metric cost matrix - - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - a and b are source and target weights + - M is the (dim_a, dim_b) metric cost matrix + - :math:`\Omega` is the entropic regularization term + :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` + - a and b are source and target unbalanced distributions - KL is the Kullback-Leibler divergence - The algorithm used for solving the problem is the generalized Sinkhorn-Knopp matrix scaling algorithm as proposed in [10, 23]_ + The algorithm used for solving the problem is the generalized + Sinkhorn-Knopp matrix scaling algorithm as proposed in [10, 23]_ Parameters ---------- - a : np.ndarray (ns,) - samples weights in the source domain - b : np.ndarray (nt,) or np.ndarray (nt, n_hists) - samples in the target domain, compute sinkhorn with multiple targets - and fixed M if b is a matrix (return OT loss + dual variables in log) - M : np.ndarray (ns,nt) + a : np.ndarray (dim_a,) + Unnormalized histogram of dimension dim_a + b : np.ndarray (dim_b,) or np.ndarray (dim_b, n_hists) + One or multiple unnormalized histograms of dimension dim_b + If many, compute all the OT distances (a, b_i) + M : np.ndarray (dim_a, dim_b) loss matrix reg : float Entropy regularization term > 0 - alpha : float + reg_m: float Marginal relaxation term > 0 method : str method used for the solver either 'sinkhorn', 'sinkhorn_stabilized' or - 'sinkhorn_epsilon_scaling', see those function for specific parameters + 'sinkhorn_reg_scaling', see those function for specific parameters numItermax : int, optional Max number of iterations stopThr : float, optional @@ -171,10 +195,10 @@ def sinkhorn_unbalanced2(a, b, M, reg, alpha, method='sinkhorn', Returns ------- - W : (nt) ndarray or float - Optimal transportation matrix for the given parameters + ot_distance : (n_hists,) ndarray + the OT distance between `a` and each of the histograms `b_i` log : dict - log dictionary return only if log==True in parameters + log dictionary returned only if `log` is `True` Examples -------- @@ -191,64 +215,70 @@ def sinkhorn_unbalanced2(a, b, M, reg, alpha, method='sinkhorn', References ---------- - .. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013 + .. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal + Transport, Advances in Neural Information Processing Systems + (NIPS) 26, 2013 - .. [9] Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519. + .. [9] Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms for + Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519. - .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816. + .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). + Scaling algorithms for unbalanced transport problems. arXiv preprint + arXiv:1607.05816. - .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss, Advances in Neural Information Processing Systems (NIPS) 2015 + .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : + Learning with a Wasserstein Loss, Advances in Neural Information + Processing Systems (NIPS) 2015 See Also -------- ot.unbalanced.sinkhorn_knopp : Unbalanced Classic Sinkhorn [10] ot.unbalanced.sinkhorn_stabilized: Unbalanced Stabilized sinkhorn [9][10] - ot.unbalanced.sinkhorn_epsilon_scaling: Unbalanced Sinkhorn with epslilon scaling [9][10] + ot.unbalanced.sinkhorn_reg_scaling: Unbalanced Sinkhorn with epslilon scaling [9][10] """ - - if method.lower() == 'sinkhorn': - def sink(): - return sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) - - elif method.lower() in ['sinkhorn_stabilized', 'sinkhorn_epsilon_scaling']: - warnings.warn('Method not implemented yet. Using classic Sinkhorn Knopp') - - def sink(): - return sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) - else: - raise ValueError('Unknown method. Using classic Sinkhorn Knopp') - b = np.asarray(b, dtype=np.float64) if len(b.shape) < 2: b = b[:, None] - - return sink() + if method.lower() == 'sinkhorn': + return _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) + + elif method.lower() == 'sinkhorn_stabilized': + return _sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, + verbose=verbose, + log=log, **kwargs) + elif method.lower() in ['sinkhorn_reg_scaling']: + warnings.warn('Method not implemented yet. Using classic Sinkhorn Knopp') + return _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) + else: + raise ValueError('Unknown method %s.' % method) -def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000, - stopThr=1e-9, verbose=False, log=False, **kwargs): +def _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, numItermax=1000, + stopThr=1e-6, verbose=False, log=False, **kwargs): r""" Solve the entropic regularization unbalanced optimal transport problem and return the loss The function solves the following optimization problem: .. math:: - W = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma) + \\alpha KL(\gamma 1, a) + \\alpha KL(\gamma^T 1, b) + W = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma) + \reg_m KL(\gamma 1, a) + \reg_m KL(\gamma^T 1, b) s.t. \gamma\geq 0 where : - - M is the (ns, nt) metric cost matrix + - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - a and b are source and target weights + - a and b are source and target unbalanced distributions - KL is the Kullback-Leibler divergence The algorithm used for solving the problem is the generalized Sinkhorn-Knopp matrix scaling algorithm as proposed in [10, 23]_ @@ -256,16 +286,16 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000, Parameters ---------- - a : np.ndarray (ns,) - samples weights in the source domain - b : np.ndarray (nt,) or np.ndarray (nt, n_hists) - samples in the target domain, compute sinkhorn with multiple targets - and fixed M if b is a matrix (return OT loss + dual variables in log) - M : np.ndarray (ns,nt) + a : np.ndarray (dim_a,) + Unnormalized histogram of dimension dim_a + b : np.ndarray (dim_b,) or np.ndarray (dim_b, n_hists) + One or multiple unnormalized histograms of dimension dim_b + If many, compute all the OT distances (a, b_i) + M : np.ndarray (dim_a, dim_b) loss matrix reg : float Entropy regularization term > 0 - alpha : float + reg_m: float Marginal relaxation term > 0 numItermax : int, optional Max number of iterations @@ -279,11 +309,16 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000, Returns ------- - gamma : (ns x nt) ndarray - Optimal transportation matrix for the given parameters - log : dict - log dictionary return only if log==True in parameters - + if n_hists == 1: + gamma : (dim_a x dim_b) ndarray + Optimal transportation matrix for the given parameters + log : dict + log dictionary returned only if `log` is `True` + else: + ot_distance : (n_hists,) ndarray + the OT distance between `a` and each of the histograms `b_i` + log : dict + log dictionary returned only if `log` is `True` Examples -------- @@ -291,16 +326,20 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000, >>> a=[.5, .5] >>> b=[.5, .5] >>> M=[[0., 1.],[1., 0.]] - >>> ot.unbalanced.sinkhorn_knopp_unbalanced(a, b, M, 1., 1.) + >>> ot.unbalanced._sinkhorn_knopp_unbalanced(a, b, M, 1., 1.) array([[0.51122823, 0.18807035], [0.18807035, 0.51122823]]) References ---------- - .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816. + .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). + Scaling algorithms for unbalanced transport problems. arXiv preprint + arXiv:1607.05816. - .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss, Advances in Neural Information Processing Systems (NIPS) 2015 + .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : + Learning with a Wasserstein Loss, Advances in Neural Information + Processing Systems (NIPS) 2015 See Also -------- @@ -313,12 +352,12 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000, b = np.asarray(b, dtype=np.float64) M = np.asarray(M, dtype=np.float64) - n_a, n_b = M.shape + dim_a, dim_b = M.shape if len(a) == 0: - a = np.ones(n_a, dtype=np.float64) / n_a + a = np.ones(dim_a, dtype=np.float64) / dim_a if len(b) == 0: - b = np.ones(n_b, dtype=np.float64) / n_b + b = np.ones(dim_b, dtype=np.float64) / dim_b if len(b.shape) > 1: n_hists = b.shape[1] @@ -331,21 +370,19 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000, # we assume that no distances are null except those of the diagonal of # distances if n_hists: - u = np.ones((n_a, 1)) / n_a - v = np.ones((n_b, n_hists)) / n_b - a = a.reshape(n_a, 1) + u = np.ones((dim_a, 1)) / dim_a + v = np.ones((dim_b, n_hists)) / dim_b + a = a.reshape(dim_a, 1) else: - u = np.ones(n_a) / n_a - v = np.ones(n_b) / n_b + u = np.ones(dim_a) / dim_a + v = np.ones(dim_b) / dim_b - # print(reg) # Next 3 lines equivalent to K= np.exp(-M/reg), but faster to compute K = np.empty(M.shape, dtype=M.dtype) np.divide(M, -reg, out=K) np.exp(K, out=K) - # print(np.min(K)) - fi = alpha / (alpha + reg) + fi = reg_m / (reg_m + reg) cpt = 0 err = 1. @@ -371,8 +408,9 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000, if cpt % 10 == 0: # we can speed up the process by checking for the error only all # the 10th iterations - err = np.sum((u - uprev)**2) / np.sum((u)**2) + \ - np.sum((v - vprev)**2) / np.sum((v)**2) + err_u = abs(u - uprev).max() / max(abs(u).max(), abs(uprev).max(), 1.) + err_v = abs(v - vprev).max() / max(abs(v).max(), abs(vprev).max(), 1.) + err = 0.5 * (err_u + err_v) if log: log['err'].append(err) if verbose: @@ -383,8 +421,8 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000, cpt += 1 if log: - log['u'] = u - log['v'] = v + log['logu'] = np.log(u + 1e-16) + log['logv'] = np.log(v + 1e-16) if n_hists: # return only loss res = np.einsum('ik,ij,jk,ij->k', u, K, v, M) @@ -401,9 +439,224 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, alpha, numItermax=1000, return u[:, None] * K * v[None, :] -def barycenter_unbalanced(A, M, reg, alpha, weights=None, numItermax=1000, - stopThr=1e-4, verbose=False, log=False): - r"""Compute the entropic regularized unbalanced wasserstein barycenter of distributions A +def _sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, tau=1e5, numItermax=1000, + stopThr=1e-6, verbose=False, log=False, + **kwargs): + r""" + Solve the entropic regularization unbalanced optimal transport + problem and return the loss + + The function solves the following optimization problem using log-domain + stabilization as proposed in [10]: + + .. math:: + W = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma) + reg_m KL(\gamma 1, a) + reg_m KL(\gamma^T 1, b) + + s.t. + \gamma\geq 0 + where : + + - M is the (dim_a, dim_b) metric cost matrix + - :math:`\Omega` is the entropic regularization + term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` + - a and b are source and target unbalanced distributions + - KL is the Kullback-Leibler divergence + + The algorithm used for solving the problem is the generalized + Sinkhorn-Knopp matrix scaling algorithm as proposed in [10, 23]_ + + + Parameters + ---------- + a : np.ndarray (dim_a,) + Unnormalized histogram of dimension dim_a + b : np.ndarray (dim_b,) or np.ndarray (dim_b, n_hists) + One or multiple unnormalized histograms of dimension dim_b + If many, compute all the OT distances (a, b_i) + M : np.ndarray (dim_a, dim_b) + loss matrix + reg : float + Entropy regularization term > 0 + reg_m: float + Marginal relaxation term > 0 + tau : float + thershold for max value in u or v for log scaling + 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 + ------- + if n_hists == 1: + gamma : (dim_a x dim_b) ndarray + Optimal transportation matrix for the given parameters + log : dict + log dictionary returned only if `log` is `True` + else: + ot_distance : (n_hists,) ndarray + the OT distance between `a` and each of the histograms `b_i` + log : dict + log dictionary returned only if `log` is `True` + Examples + -------- + + >>> import ot + >>> a=[.5, .5] + >>> b=[.5, .5] + >>> M=[[0., 1.],[1., 0.]] + >>> ot.unbalanced._sinkhorn_stabilized_unbalanced(a, b, M, 1., 1.) + array([[0.51122823, 0.18807035], + [0.18807035, 0.51122823]]) + + References + ---------- + + .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). + Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816. + + .. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : + Learning with a Wasserstein Loss, Advances in Neural Information + Processing Systems (NIPS) 2015 + + See Also + -------- + ot.lp.emd : Unregularized OT + ot.optim.cg : General regularized OT + + """ + + a = np.asarray(a, dtype=np.float64) + b = np.asarray(b, dtype=np.float64) + M = np.asarray(M, dtype=np.float64) + + dim_a, dim_b = M.shape + + if len(a) == 0: + a = np.ones(dim_a, dtype=np.float64) / dim_a + if len(b) == 0: + b = np.ones(dim_b, dtype=np.float64) / dim_b + + if len(b.shape) > 1: + n_hists = b.shape[1] + else: + n_hists = 0 + + if log: + log = {'err': []} + + # we assume that no distances are null except those of the diagonal of + # distances + if n_hists: + u = np.ones((dim_a, n_hists)) / dim_a + v = np.ones((dim_b, n_hists)) / dim_b + a = a.reshape(dim_a, 1) + else: + u = np.ones(dim_a) / dim_a + v = np.ones(dim_b) / dim_b + + # print(reg) + # Next 3 lines equivalent to K= np.exp(-M/reg), but faster to compute + K = np.empty(M.shape, dtype=M.dtype) + np.divide(M, -reg, out=K) + np.exp(K, out=K) + + fi = reg_m / (reg_m + reg) + + cpt = 0 + err = 1. + alpha = np.zeros(dim_a) + beta = np.zeros(dim_b) + while (err > stopThr and cpt < numItermax): + uprev = u + vprev = v + + Kv = K.dot(v) + f_alpha = np.exp(- alpha / (reg + reg_m)) + f_beta = np.exp(- beta / (reg + reg_m)) + + if n_hists: + f_alpha = f_alpha[:, None] + f_beta = f_beta[:, None] + u = ((a / (Kv + 1e-16)) ** fi) * f_alpha + Ktu = K.T.dot(u) + v = ((b / (Ktu + 1e-16)) ** fi) * f_beta + absorbing = False + if (u > tau).any() or (v > tau).any(): + absorbing = True + if n_hists: + alpha = alpha + reg * np.log(np.max(u, 1)) + beta = beta + reg * np.log(np.max(v, 1)) + else: + alpha = alpha + reg * np.log(np.max(u)) + beta = beta + reg * np.log(np.max(v)) + K = np.exp((alpha[:, None] + beta[None, :] - + M) / reg) + v = np.ones_like(v) + Kv = K.dot(v) + + if (np.any(Ktu == 0.) + or np.any(np.isnan(u)) or np.any(np.isnan(v)) + or np.any(np.isinf(u)) or np.any(np.isinf(v))): + # we have reached the machine precision + # come back to previous solution and quit loop + warnings.warn('Numerical errors at iteration %s' % cpt) + u = uprev + v = vprev + break + if (cpt % 10 == 0 and not absorbing) or cpt == 0: + # we can speed up the process by checking for the error only all + # the 10th iterations + err = abs(u - uprev).max() / max(abs(u).max(), abs(uprev).max(), + 1.) + if log: + log['err'].append(err) + if verbose: + if cpt % 200 == 0: + print( + '{:5s}|{:12s}'.format('It.', 'Err') + '\n' + '-' * 19) + print('{:5d}|{:8e}|'.format(cpt, err)) + cpt = cpt + 1 + + if err > stopThr: + warnings.warn("Stabilized Unbalanced Sinkhorn did not converge." + + "Try a larger entropy `reg` or a lower mass `reg_m`." + + "Or a larger absorption threshold `tau`.") + if n_hists: + logu = alpha[:, None] / reg + np.log(u) + logv = beta[:, None] / reg + np.log(v) + else: + logu = alpha / reg + np.log(u) + logv = beta / reg + np.log(v) + if log: + log['logu'] = logu + log['logv'] = logv + if n_hists: # return only loss + res = logsumexp(np.log(M + 1e-100)[:, :, None] + logu[:, None, :] + + logv[None, :, :] - M[:, :, None] / reg, axis=(0, 1)) + res = np.exp(res) + if log: + return res, log + else: + return res + + else: # return OT matrix + ot_matrix = np.exp(logu[:, None] + logv[None, :] - M / reg) + if log: + return ot_matrix, log + else: + return ot_matrix + + +def _barycenter_unbalanced_stabilized(A, M, reg, reg_m, weights=None, tau=1e3, + numItermax=1000, stopThr=1e-6, + verbose=False, log=False): + r"""Compute the entropic unbalanced wasserstein barycenter of A with stabilization. The function solves the following optimization problem: @@ -412,28 +665,184 @@ def barycenter_unbalanced(A, M, reg, alpha, weights=None, numItermax=1000, where : - - :math:`Wu_{reg}(\cdot,\cdot)` is the unbalanced entropic regularized Wasserstein distance (see ot.unbalanced.sinkhorn_unbalanced) - - :math:`\mathbf{a}_i` are training distributions in the columns of matrix :math:`\mathbf{A}` - - reg and :math:`\mathbf{M}` are respectively the regularization term and the cost matrix for OT - - alpha is the marginal relaxation hyperparameter - The algorithm used for solving the problem is the generalized Sinkhorn-Knopp matrix scaling algorithm as proposed in [10]_ + - :math:`Wu_{reg}(\cdot,\cdot)` is the unbalanced entropic regularized + Wasserstein distance (see ot.unbalanced.sinkhorn_unbalanced) + - :math:`\mathbf{a}_i` are training distributions in the columns of + matrix :math:`\mathbf{A}` + - reg and :math:`\mathbf{M}` are respectively the regularization term and + the cost matrix for OT + - reg_mis the marginal relaxation hyperparameter + The algorithm used for solving the problem is the generalized + Sinkhorn-Knopp matrix scaling algorithm as proposed in [10]_ Parameters ---------- - A : np.ndarray (d,n) - n training distributions a_i of size d - M : np.ndarray (d,d) - loss matrix for OT + A : np.ndarray (dim, n_hists) + `n_hists` training distributions a_i of dimension dim + M : np.ndarray (dim, dim) + ground metric matrix for OT. reg : float Entropy regularization term > 0 - alpha : float + reg_m : float Marginal relaxation term > 0 - weights : np.ndarray (n,) - Weights of each histogram a_i on the simplex (barycentric coodinates) + tau : float + Stabilization threshold for log domain absorption. + weights : np.ndarray (n_hists,) optional + Weight of each distribution (barycentric coodinates) + If None, uniform weights are used. numItermax : int, optional Max number of iterations stopThr : float, optional - Stop threshol on error (>0) + Stop threshol on error (> 0) + verbose : bool, optional + Print information along iterations + log : bool, optional + record log if True + + + Returns + ------- + a : (dim,) ndarray + Unbalanced Wasserstein barycenter + log : dict + log dictionary return only if log==True in parameters + + + References + ---------- + + .. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, + G. (2015). Iterative Bregman projections for regularized transportation + problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138. + .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). + Scaling algorithms for unbalanced transport problems. arXiv preprint + arXiv:1607.05816. + + + """ + dim, n_hists = A.shape + if weights is None: + weights = np.ones(n_hists) / n_hists + else: + assert(len(weights) == A.shape[1]) + + if log: + log = {'err': []} + + fi = reg_m / (reg_m + reg) + + u = np.ones((dim, n_hists)) / dim + v = np.ones((dim, n_hists)) / dim + + # print(reg) + # Next 3 lines equivalent to K= np.exp(-M/reg), but faster to compute + K = np.empty(M.shape, dtype=M.dtype) + np.divide(M, -reg, out=K) + np.exp(K, out=K) + + fi = reg_m / (reg_m + reg) + + cpt = 0 + err = 1. + alpha = np.zeros(dim) + beta = np.zeros(dim) + q = np.ones(dim) / dim + while (err > stopThr and cpt < numItermax): + qprev = q + Kv = K.dot(v) + f_alpha = np.exp(- alpha / (reg + reg_m)) + f_beta = np.exp(- beta / (reg + reg_m)) + f_alpha = f_alpha[:, None] + f_beta = f_beta[:, None] + u = ((A / (Kv + 1e-16)) ** fi) * f_alpha + Ktu = K.T.dot(u) + q = (Ktu ** (1 - fi)) * f_beta + q = q.dot(weights) ** (1 / (1 - fi)) + Q = q[:, None] + v = ((Q / (Ktu + 1e-16)) ** fi) * f_beta + absorbing = False + if (u > tau).any() or (v > tau).any(): + absorbing = True + alpha = alpha + reg * np.log(np.max(u, 1)) + beta = beta + reg * np.log(np.max(v, 1)) + K = np.exp((alpha[:, None] + beta[None, :] - + M) / reg) + v = np.ones_like(v) + Kv = K.dot(v) + if (np.any(Ktu == 0.) + or np.any(np.isnan(u)) or np.any(np.isnan(v)) + or np.any(np.isinf(u)) or np.any(np.isinf(v))): + # we have reached the machine precision + # come back to previous solution and quit loop + warnings.warn('Numerical errors at iteration %s' % cpt) + q = qprev + break + if (cpt % 10 == 0 and not absorbing) or cpt == 0: + # we can speed up the process by checking for the error only all + # the 10th iterations + err = abs(q - qprev).max() / max(abs(q).max(), + abs(qprev).max(), 1.) + if log: + log['err'].append(err) + if verbose: + if cpt % 50 == 0: + print( + '{:5s}|{:12s}'.format('It.', 'Err') + '\n' + '-' * 19) + print('{:5d}|{:8e}|'.format(cpt, err)) + + cpt += 1 + if err > stopThr: + warnings.warn("Stabilized Unbalanced Sinkhorn did not converge." + + "Try a larger entropy `reg` or a lower mass `reg_m`." + + "Or a larger absorption threshold `tau`.") + if log: + log['niter'] = cpt + log['logu'] = np.log(u + 1e-16) + log['logv'] = np.log(v + 1e-16) + return q, log + else: + return q + + +def _barycenter_unbalanced(A, M, reg, reg_m, weights=None, + numItermax=1000, stopThr=1e-6, + verbose=False, log=False): + r"""Compute the entropic unbalanced wasserstein barycenter of A. + + The function solves the following optimization problem with a + + .. math:: + \mathbf{a} = arg\min_\mathbf{a} \sum_i Wu_{reg}(\mathbf{a},\mathbf{a}_i) + + where : + + - :math:`Wu_{reg}(\cdot,\cdot)` is the unbalanced entropic regularized + Wasserstein distance (see ot.unbalanced.sinkhorn_unbalanced) + - :math:`\mathbf{a}_i` are training distributions in the columns of matrix + :math:`\mathbf{A}` + - reg and :math:`\mathbf{M}` are respectively the regularization term and + the cost matrix for OT + - reg_mis the marginal relaxation hyperparameter + The algorithm used for solving the problem is the generalized + Sinkhorn-Knopp matrix scaling algorithm as proposed in [10]_ + + Parameters + ---------- + A : np.ndarray (dim, n_hists) + `n_hists` training distributions a_i of dimension dim + M : np.ndarray (dim, dim) + ground metric matrix for OT. + reg : float + Entropy regularization term > 0 + reg_m: float + Marginal relaxation term > 0 + weights : np.ndarray (n_hists,) optional + Weight of each distribution (barycentric coodinates) + If None, uniform weights are used. + 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 @@ -442,7 +851,7 @@ def barycenter_unbalanced(A, M, reg, alpha, weights=None, numItermax=1000, Returns ------- - a : (d,) ndarray + a : (dim,) ndarray Unbalanced Wasserstein barycenter log : dict log dictionary return only if log==True in parameters @@ -451,12 +860,16 @@ def barycenter_unbalanced(A, M, reg, alpha, weights=None, numItermax=1000, References ---------- - .. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015). Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138. - .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816. + .. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. + (2015). Iterative Bregman projections for regularized transportation + problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138. + .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). + Scaling algorithms for unbalanced transport problems. arXiv preprin + arXiv:1607.05816. """ - p, n_hists = A.shape + dim, n_hists = A.shape if weights is None: weights = np.ones(n_hists) / n_hists else: @@ -467,10 +880,10 @@ def barycenter_unbalanced(A, M, reg, alpha, weights=None, numItermax=1000, K = np.exp(- M / reg) - fi = alpha / (alpha + reg) + fi = reg_m / (reg_m + reg) - v = np.ones((p, n_hists)) / p - u = np.ones((p, 1)) / p + v = np.ones((dim, n_hists)) / dim + u = np.ones((dim, 1)) / dim cpt = 0 err = 1. @@ -499,8 +912,11 @@ def barycenter_unbalanced(A, M, reg, alpha, weights=None, numItermax=1000, if cpt % 10 == 0: # we can speed up the process by checking for the error only all # the 10th iterations - err = np.sum((u - uprev) ** 2) / np.sum((u) ** 2) + \ - np.sum((v - vprev) ** 2) / np.sum((v) ** 2) + err_u = abs(u - uprev).max() + err_u /= max(abs(u).max(), abs(uprev).max(), 1.) + err_v = abs(v - vprev).max() + err_v /= max(abs(v).max(), abs(vprev).max(), 1.) + err = 0.5 * (err_u + err_v) if log: log['err'].append(err) if verbose: @@ -512,8 +928,95 @@ def barycenter_unbalanced(A, M, reg, alpha, weights=None, numItermax=1000, cpt += 1 if log: log['niter'] = cpt - log['u'] = u - log['v'] = v + log['logu'] = np.log(u + 1e-16) + log['logv'] = np.log(v + 1e-16) return q, log else: return q + + +def barycenter_unbalanced(A, M, reg, reg_m, method="sinkhorn", weights=None, + numItermax=1000, stopThr=1e-6, + verbose=False, log=False, **kwargs): + r"""Compute the entropic unbalanced wasserstein barycenter of A. + + The function solves the following optimization problem with a + + .. math:: + \mathbf{a} = arg\min_\mathbf{a} \sum_i Wu_{reg}(\mathbf{a},\mathbf{a}_i) + + where : + + - :math:`Wu_{reg}(\cdot,\cdot)` is the unbalanced entropic regularized + Wasserstein distance (see ot.unbalanced.sinkhorn_unbalanced) + - :math:`\mathbf{a}_i` are training distributions in the columns of matrix + :math:`\mathbf{A}` + - reg and :math:`\mathbf{M}` are respectively the regularization term and + the cost matrix for OT + - reg_mis the marginal relaxation hyperparameter + The algorithm used for solving the problem is the generalized + Sinkhorn-Knopp matrix scaling algorithm as proposed in [10]_ + + Parameters + ---------- + A : np.ndarray (dim, n_hists) + `n_hists` training distributions a_i of dimension dim + M : np.ndarray (dim, dim) + ground metric matrix for OT. + reg : float + Entropy regularization term > 0 + reg_m: float + Marginal relaxation term > 0 + weights : np.ndarray (n_hists,) optional + Weight of each distribution (barycentric coodinates) + If None, uniform weights are used. + 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 + ------- + a : (dim,) ndarray + Unbalanced Wasserstein barycenter + log : dict + log dictionary return only if log==True in parameters + + + References + ---------- + + .. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. + (2015). Iterative Bregman projections for regularized transportation + problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138. + .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). + Scaling algorithms for unbalanced transport problems. arXiv preprin + arXiv:1607.05816. + + """ + + if method.lower() == 'sinkhorn': + return _barycenter_unbalanced(A, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) + + elif method.lower() == 'sinkhorn_stabilized': + return _barycenter_unbalanced_stabilized(A, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, + verbose=verbose, + log=log, **kwargs) + elif method.lower() in ['sinkhorn_reg_scaling']: + warnings.warn('Method not implemented yet. Using classic Sinkhorn Knopp') + return _barycenter_unbalanced(A, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) + else: + raise ValueError("Unknown method '%s'." % method) diff --git a/pytest.ini b/pytest.ini new file mode 100644 index 0000000..e69de29 diff --git a/test/test_bregman.py b/test/test_bregman.py index 83ebba8..f70df10 100644 --- a/test/test_bregman.py +++ b/test/test_bregman.py @@ -7,6 +7,7 @@ import numpy as np import ot +import pytest def test_sinkhorn(): @@ -71,13 +72,11 @@ def test_sinkhorn_variants(): Gs = ot.sinkhorn(u, u, M, 1, method='sinkhorn_stabilized', stopThr=1e-10) Ges = ot.sinkhorn( u, u, M, 1, method='sinkhorn_epsilon_scaling', stopThr=1e-10) - Gerr = ot.sinkhorn(u, u, M, 1, method='do_not_exists', stopThr=1e-10) G_green = ot.sinkhorn(u, u, M, 1, method='greenkhorn', stopThr=1e-10) # check values np.testing.assert_allclose(G0, Gs, atol=1e-05) np.testing.assert_allclose(G0, Ges, atol=1e-05) - np.testing.assert_allclose(G0, Gerr) np.testing.assert_allclose(G0, G_green, atol=1e-5) print(G0, G_green) @@ -96,18 +95,17 @@ def test_sinkhorn_variants_log(): Gs, logs = ot.sinkhorn(u, u, M, 1, method='sinkhorn_stabilized', stopThr=1e-10, log=True) Ges, loges = ot.sinkhorn( u, u, M, 1, method='sinkhorn_epsilon_scaling', stopThr=1e-10, log=True) - Gerr, logerr = ot.sinkhorn(u, u, M, 1, method='do_not_exists', stopThr=1e-10, log=True) G_green, loggreen = ot.sinkhorn(u, u, M, 1, method='greenkhorn', stopThr=1e-10, log=True) # check values np.testing.assert_allclose(G0, Gs, atol=1e-05) np.testing.assert_allclose(G0, Ges, atol=1e-05) - np.testing.assert_allclose(G0, Gerr) np.testing.assert_allclose(G0, G_green, atol=1e-5) print(G0, G_green) -def test_bary(): +@pytest.mark.parametrize("method", ["sinkhorn", "sinkhorn_stabilized"]) +def test_barycenter(method): n_bins = 100 # nb bins @@ -126,14 +124,42 @@ def test_bary(): weights = np.array([1 - alpha, alpha]) # wasserstein - reg = 1e-3 - bary_wass = ot.bregman.barycenter(A, M, reg, weights) + reg = 1e-2 + bary_wass = ot.bregman.barycenter(A, M, reg, weights, method=method) np.testing.assert_allclose(1, np.sum(bary_wass)) ot.bregman.barycenter(A, M, reg, log=True, verbose=True) +def test_barycenter_stabilization(): + + n_bins = 100 # nb bins + + # Gaussian distributions + a1 = ot.datasets.make_1D_gauss(n_bins, m=30, s=10) # m= mean, s= std + a2 = ot.datasets.make_1D_gauss(n_bins, m=40, s=10) + + # creating matrix A containing all distributions + A = np.vstack((a1, a2)).T + + # loss matrix + normalization + M = ot.utils.dist0(n_bins) + M /= M.max() + + alpha = 0.5 # 0<=alpha<=1 + weights = np.array([1 - alpha, alpha]) + + # wasserstein + reg = 1e-2 + bar_stable = ot.bregman.barycenter(A, M, reg, weights, + method="sinkhorn_stabilized", + stopThr=1e-8) + bar = ot.bregman.barycenter(A, M, reg, weights, method="sinkhorn", + stopThr=1e-8) + np.testing.assert_allclose(bar, bar_stable) + + def test_wasserstein_bary_2d(): size = 100 # size of a square image @@ -279,3 +305,35 @@ def test_stabilized_vs_sinkhorn_multidim(): method="sinkhorn", log=True) np.testing.assert_allclose(G, G2) + + +def test_implemented_methods(): + IMPLEMENTED_METHODS = ['sinkhorn', 'sinkhorn_stabilized'] + ONLY_1D_methods = ['greenkhorn', 'sinkhorn_epsilon_scaling'] + NOT_VALID_TOKENS = ['foo'] + # test generalized sinkhorn for unbalanced OT barycenter + n = 3 + rng = np.random.RandomState(42) + + x = rng.randn(n, 2) + a = ot.utils.unif(n) + + # make dists unbalanced + b = ot.utils.unif(n) + A = rng.rand(n, 2) + M = ot.dist(x, x) + epsilon = 1. + + for method in IMPLEMENTED_METHODS: + ot.bregman.sinkhorn(a, b, M, epsilon, method=method) + ot.bregman.sinkhorn2(a, b, M, epsilon, method=method) + ot.bregman.barycenter(A, M, reg=epsilon, method=method) + with pytest.raises(ValueError): + for method in set(NOT_VALID_TOKENS): + ot.bregman.sinkhorn(a, b, M, epsilon, method=method) + ot.bregman.sinkhorn2(a, b, M, epsilon, method=method) + ot.bregman.barycenter(A, M, reg=epsilon, method=method) + for method in ONLY_1D_methods: + ot.bregman.sinkhorn(a, b, M, epsilon, method=method) + with pytest.raises(ValueError): + ot.bregman.sinkhorn2(a, b, M, epsilon, method=method) diff --git a/test/test_unbalanced.py b/test/test_unbalanced.py index 1395fe1..ca1efba 100644 --- a/test/test_unbalanced.py +++ b/test/test_unbalanced.py @@ -7,9 +7,12 @@ import numpy as np import ot import pytest +from ot.unbalanced import barycenter_unbalanced +from scipy.special import logsumexp -@pytest.mark.parametrize("method", ["sinkhorn"]) + +@pytest.mark.parametrize("method", ["sinkhorn", "sinkhorn_stabilized"]) def test_unbalanced_convergence(method): # test generalized sinkhorn for unbalanced OT n = 100 @@ -23,29 +26,35 @@ def test_unbalanced_convergence(method): M = ot.dist(x, x) epsilon = 1. - alpha = 1. - K = np.exp(- M / epsilon) + reg_m = 1. - G, log = ot.unbalanced.sinkhorn_unbalanced(a, b, M, reg=epsilon, alpha=alpha, - stopThr=1e-10, method=method, + G, log = ot.unbalanced.sinkhorn_unbalanced(a, b, M, reg=epsilon, + reg_m=reg_m, + method=method, log=True) - loss = ot.unbalanced.sinkhorn_unbalanced2(a, b, M, epsilon, alpha, + loss = ot.unbalanced.sinkhorn_unbalanced2(a, b, M, epsilon, reg_m, method=method) # check fixed point equations - fi = alpha / (alpha + epsilon) - v_final = (b / K.T.dot(log["u"])) ** fi - u_final = (a / K.dot(log["v"])) ** fi + # in log-domain + fi = reg_m / (reg_m + epsilon) + logb = np.log(b + 1e-16) + loga = np.log(a + 1e-16) + logKtu = logsumexp(log["logu"][None, :] - M.T / epsilon, axis=1) + logKv = logsumexp(log["logv"][None, :] - M / epsilon, axis=1) + + v_final = fi * (logb - logKtu) + u_final = fi * (loga - logKv) np.testing.assert_allclose( - u_final, log["u"], atol=1e-05) + u_final, log["logu"], atol=1e-05) np.testing.assert_allclose( - v_final, log["v"], atol=1e-05) + v_final, log["logv"], atol=1e-05) # check if sinkhorn_unbalanced2 returns the correct loss np.testing.assert_allclose((G * M).sum(), loss, atol=1e-5) -@pytest.mark.parametrize("method", ["sinkhorn"]) +@pytest.mark.parametrize("method", ["sinkhorn", "sinkhorn_stabilized"]) def test_unbalanced_multiple_inputs(method): # test generalized sinkhorn for unbalanced OT n = 100 @@ -59,28 +68,59 @@ def test_unbalanced_multiple_inputs(method): M = ot.dist(x, x) epsilon = 1. - alpha = 1. - K = np.exp(- M / epsilon) + reg_m = 1. loss, log = ot.unbalanced.sinkhorn_unbalanced(a, b, M, reg=epsilon, - alpha=alpha, - stopThr=1e-10, method=method, + reg_m=reg_m, + method=method, log=True) # check fixed point equations - fi = alpha / (alpha + epsilon) - v_final = (b / K.T.dot(log["u"])) ** fi - - u_final = (a[:, None] / K.dot(log["v"])) ** fi + # in log-domain + fi = reg_m / (reg_m + epsilon) + logb = np.log(b + 1e-16) + loga = np.log(a + 1e-16)[:, None] + logKtu = logsumexp(log["logu"][:, None, :] - M[:, :, None] / epsilon, + axis=0) + logKv = logsumexp(log["logv"][None, :] - M[:, :, None] / epsilon, axis=1) + v_final = fi * (logb - logKtu) + u_final = fi * (loga - logKv) np.testing.assert_allclose( - u_final, log["u"], atol=1e-05) + u_final, log["logu"], atol=1e-05) np.testing.assert_allclose( - v_final, log["v"], atol=1e-05) + v_final, log["logv"], atol=1e-05) assert len(loss) == b.shape[1] -def test_unbalanced_barycenter(): +def test_stabilized_vs_sinkhorn(): + # test if stable version matches sinkhorn + n = 100 + + # Gaussian distributions + a = ot.datasets.make_1D_gauss(n, m=20, s=5) # m= mean, s= std + b1 = ot.datasets.make_1D_gauss(n, m=60, s=8) + b2 = ot.datasets.make_1D_gauss(n, m=30, s=4) + + # creating matrix A containing all distributions + b = np.vstack((b1, b2)).T + + M = ot.utils.dist0(n) + M /= np.median(M) + epsilon = 0.1 + reg_m = 1. + G, log = ot.unbalanced.sinkhorn_unbalanced2(a, b, M, reg=epsilon, + method="sinkhorn_stabilized", + reg_m=reg_m, + log=True) + G2, log2 = ot.unbalanced.sinkhorn_unbalanced2(a, b, M, epsilon, reg_m, + method="sinkhorn", log=True) + + np.testing.assert_allclose(G, G2, atol=1e-5) + + +@pytest.mark.parametrize("method", ["sinkhorn", "sinkhorn_stabilized"]) +def test_unbalanced_barycenter(method): # test generalized sinkhorn for unbalanced OT barycenter n = 100 rng = np.random.RandomState(42) @@ -92,27 +132,56 @@ def test_unbalanced_barycenter(): A = A * np.array([1, 2])[None, :] M = ot.dist(x, x) epsilon = 1. - alpha = 1. - K = np.exp(- M / epsilon) + reg_m = 1. - q, log = ot.unbalanced.barycenter_unbalanced(A, M, reg=epsilon, alpha=alpha, - stopThr=1e-10, - log=True) + q, log = barycenter_unbalanced(A, M, reg=epsilon, reg_m=reg_m, + method=method, log=True) # check fixed point equations - fi = alpha / (alpha + epsilon) - v_final = (q[:, None] / K.T.dot(log["u"])) ** fi - u_final = (A / K.dot(log["v"])) ** fi + fi = reg_m / (reg_m + epsilon) + logA = np.log(A + 1e-16) + logq = np.log(q + 1e-16)[:, None] + logKtu = logsumexp(log["logu"][:, None, :] - M[:, :, None] / epsilon, + axis=0) + logKv = logsumexp(log["logv"][None, :] - M[:, :, None] / epsilon, axis=1) + v_final = fi * (logq - logKtu) + u_final = fi * (logA - logKv) np.testing.assert_allclose( - u_final, log["u"], atol=1e-05) + u_final, log["logu"], atol=1e-05) np.testing.assert_allclose( - v_final, log["v"], atol=1e-05) + v_final, log["logv"], atol=1e-05) + + +def test_barycenter_stabilized_vs_sinkhorn(): + # test generalized sinkhorn for unbalanced OT barycenter + n = 100 + rng = np.random.RandomState(42) + + x = rng.randn(n, 2) + A = rng.rand(n, 2) + + # make dists unbalanced + A = A * np.array([1, 4])[None, :] + M = ot.dist(x, x) + epsilon = 0.5 + reg_m = 10 + + qstable, log = barycenter_unbalanced(A, M, reg=epsilon, + reg_m=reg_m, log=True, + tau=100, + method="sinkhorn_stabilized", + ) + q, log = barycenter_unbalanced(A, M, reg=epsilon, reg_m=reg_m, + method="sinkhorn", + log=True) + + np.testing.assert_allclose( + q, qstable, atol=1e-05) def test_implemented_methods(): - IMPLEMENTED_METHODS = ['sinkhorn'] - TO_BE_IMPLEMENTED_METHODS = ['sinkhorn_stabilized', - 'sinkhorn_epsilon_scaling'] + IMPLEMENTED_METHODS = ['sinkhorn', 'sinkhorn_stabilized'] + TO_BE_IMPLEMENTED_METHODS = ['sinkhorn_reg_scaling'] NOT_VALID_TOKENS = ['foo'] # test generalized sinkhorn for unbalanced OT barycenter n = 3 @@ -123,24 +192,30 @@ def test_implemented_methods(): # make dists unbalanced b = ot.utils.unif(n) * 1.5 - + A = rng.rand(n, 2) M = ot.dist(x, x) epsilon = 1. - alpha = 1. + reg_m = 1. for method in IMPLEMENTED_METHODS: - ot.unbalanced.sinkhorn_unbalanced(a, b, M, epsilon, alpha, + ot.unbalanced.sinkhorn_unbalanced(a, b, M, epsilon, reg_m, method=method) - ot.unbalanced.sinkhorn_unbalanced2(a, b, M, epsilon, alpha, + ot.unbalanced.sinkhorn_unbalanced2(a, b, M, epsilon, reg_m, method=method) + barycenter_unbalanced(A, M, reg=epsilon, reg_m=reg_m, + method=method) with pytest.warns(UserWarning, match='not implemented'): for method in set(TO_BE_IMPLEMENTED_METHODS): - ot.unbalanced.sinkhorn_unbalanced(a, b, M, epsilon, alpha, + ot.unbalanced.sinkhorn_unbalanced(a, b, M, epsilon, reg_m, method=method) - ot.unbalanced.sinkhorn_unbalanced2(a, b, M, epsilon, alpha, + ot.unbalanced.sinkhorn_unbalanced2(a, b, M, epsilon, reg_m, method=method) + barycenter_unbalanced(A, M, reg=epsilon, reg_m=reg_m, + method=method) with pytest.raises(ValueError): for method in set(NOT_VALID_TOKENS): - ot.unbalanced.sinkhorn_unbalanced(a, b, M, epsilon, alpha, + ot.unbalanced.sinkhorn_unbalanced(a, b, M, epsilon, reg_m, method=method) - ot.unbalanced.sinkhorn_unbalanced2(a, b, M, epsilon, alpha, + ot.unbalanced.sinkhorn_unbalanced2(a, b, M, epsilon, reg_m, method=method) + barycenter_unbalanced(A, M, reg=epsilon, reg_m=reg_m, + method=method) -- cgit v1.2.3 From 7f4af32c70452ff12fa8c15dea317c14d3d97878 Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Wed, 28 Aug 2019 15:40:40 +0200 Subject: correct call of unbalanced with reg_m --- ot/da.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ot/da.py b/ot/da.py index 2af855d..108a38d 100644 --- a/ot/da.py +++ b/ot/da.py @@ -1902,7 +1902,7 @@ class UnbalancedSinkhornTransport(BaseTransport): returned_ = sinkhorn_unbalanced( a=self.mu_s, b=self.mu_t, M=self.cost_, - reg=self.reg_e, alpha=self.reg_m, method=self.method, + reg=self.reg_e, reg_m=self.reg_m, method=self.method, numItermax=self.max_iter, stopThr=self.tol, verbose=self.verbose, log=self.log) -- cgit v1.2.3 From f4e8523c92a96d061040e3f25037e129d67a2d94 Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Wed, 28 Aug 2019 15:47:52 +0200 Subject: fix empirical sinkhorn doc-example --- ot/bregman.py | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/ot/bregman.py b/ot/bregman.py index 2f27d58..76698c2 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -1534,11 +1534,11 @@ def empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numI Examples -------- - >>> n_a = 2 - >>> n_b = 2 + >>> n_samples_a = 2 + >>> n_samples_b = 2 >>> reg = 0.1 - >>> X_s = np.reshape(np.arange(n_a), (dim_a, 1)) - >>> X_t = np.reshape(np.arange(0, n_b), (dim_b, 1)) + >>> X_s = np.reshape(np.arange(n_samples_a), (n_samples_a, 1)) + >>> X_t = np.reshape(np.arange(0, n_samples_b), (n_samples_b, 1)) >>> empirical_sinkhorn(X_s, X_t, reg, verbose=False) # doctest: +NORMALIZE_WHITESPACE array([[4.99977301e-01, 2.26989344e-05], [2.26989344e-05, 4.99977301e-01]]) @@ -1624,8 +1624,8 @@ def empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', num Examples -------- - >>> n_a = 2 - >>> n_b = 2 + >>> n_samples_a = 2 + >>> n_samples_b = 2 >>> reg = 0.1 >>> X_s = np.reshape(np.arange(n_samples_a), (n_samples_a, 1)) >>> X_t = np.reshape(np.arange(0, n_samples_b), (n_samples_b, 1)) @@ -1730,8 +1730,8 @@ def empirical_sinkhorn_divergence(X_s, X_t, reg, a=None, b=None, metric='sqeucli Examples -------- - >>> n_a = 2 - >>> n_b = 4 + >>> n_samples_a = 2 + >>> n_samples_b = 4 >>> reg = 0.1 >>> X_s = np.reshape(np.arange(n_samples_a), (n_samples_a, 1)) >>> X_t = np.reshape(np.arange(0, n_samples_b), (n_samples_b, 1)) -- cgit v1.2.3 From c7269d3fc72c679711699a9df7b5670b0dd176b0 Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Tue, 3 Sep 2019 17:26:22 +0200 Subject: style + make funcs public --- ot/bregman.py | 180 +++++++++++++++++++++++++++++++--------------------------- 1 file changed, 95 insertions(+), 85 deletions(-) diff --git a/ot/bregman.py b/ot/bregman.py index 76698c2..02aeb6d 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -33,7 +33,7 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, \gamma\geq 0 where : - - M is the (dim_a, n_b) metric cost matrix + - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -47,7 +47,7 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, b : ndarray, shape (dim_b,) or ndarray, shape (dim_b, n_hists) samples in the target domain, compute sinkhorn with multiple targets and fixed M if b is a matrix (return OT loss + dual variables in log) - M : ndarray, shape (dim_a, n_b) + M : ndarray, shape (dim_a, dim_b) loss matrix reg : float Regularization term >0 @@ -66,7 +66,7 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, Returns ------- - gamma : ndarray, shape (dim_a, n_b) + gamma : ndarray, shape (dim_a, dim_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -105,21 +105,21 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, """ if method.lower() == 'sinkhorn': - return _sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, - **kwargs) + return sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log, + **kwargs) elif method.lower() == 'greenkhorn': - return _greenkhorn(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log) + return greenkhorn(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log) elif method.lower() == 'sinkhorn_stabilized': - return _sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) elif method.lower() == 'sinkhorn_epsilon_scaling': - return _sinkhorn_epsilon_scaling(a, b, M, reg, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return sinkhorn_epsilon_scaling(a, b, M, reg, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) else: raise ValueError("Unknown method '%s'." % method) @@ -141,7 +141,7 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, \gamma\geq 0 where : - - M is the (dim_a, n_b) metric cost matrix + - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -155,7 +155,7 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, b : ndarray, shape (dim_b,) or ndarray, shape (dim_b, n_hists) samples in the target domain, compute sinkhorn with multiple targets and fixed M if b is a matrix (return OT loss + dual variables in log) - M : ndarray, shape (dim_a, n_b) + M : ndarray, shape (dim_a, dim_b) loss matrix reg : float Regularization term >0 @@ -173,8 +173,8 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, Returns ------- - W : (nt) ndarray or float - Optimal transportation matrix for the given parameters + W : (n_hists) ndarray or float + Optimal transportation loss for the given parameters log : dict log dictionary return only if log==True in parameters @@ -217,21 +217,23 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, if len(b.shape) < 2: b = b[:, None] if method.lower() == 'sinkhorn': - return _sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, **kwargs) + return sinkhorn_knopp(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log, + **kwargs) elif method.lower() == 'sinkhorn_stabilized': - return _sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, **kwargs) + return sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log, + **kwargs) elif method.lower() == 'sinkhorn_epsilon_scaling': - return _sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) else: raise ValueError("Unknown method '%s'." % method) -def _sinkhorn_knopp(a, b, M, reg, numItermax=1000, - stopThr=1e-9, verbose=False, log=False, **kwargs): +def sinkhorn_knopp(a, b, M, reg, numItermax=1000, + stopThr=1e-9, verbose=False, log=False, **kwargs): r""" Solve the entropic regularization optimal transport problem and return the OT matrix @@ -247,7 +249,7 @@ def _sinkhorn_knopp(a, b, M, reg, numItermax=1000, \gamma\geq 0 where : - - M is the (dim_a, n_b) metric cost matrix + - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -261,7 +263,7 @@ def _sinkhorn_knopp(a, b, M, reg, numItermax=1000, b : ndarray, shape (dim_b,) or ndarray, shape (dim_b, n_hists) samples in the target domain, compute sinkhorn with multiple targets and fixed M if b is a matrix (return OT loss + dual variables in log) - M : ndarray, shape (dim_a, n_b) + M : ndarray, shape (dim_a, dim_b) loss matrix reg : float Regularization term >0 @@ -276,7 +278,7 @@ def _sinkhorn_knopp(a, b, M, reg, numItermax=1000, Returns ------- - gamma : ndarray, shape (dim_a, n_b) + gamma : ndarray, shape (dim_a, dim_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -403,7 +405,8 @@ def _sinkhorn_knopp(a, b, M, reg, numItermax=1000, return u.reshape((-1, 1)) * K * v.reshape((1, -1)) -def _greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log=False): +def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, + log=False): r""" Solve the entropic regularization optimal transport problem and return the OT matrix @@ -427,7 +430,7 @@ def _greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log \gamma\geq 0 where : - - M is the (dim_a, n_b) metric cost matrix + - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -440,7 +443,7 @@ def _greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log b : ndarray, shape (dim_b,) or ndarray, shape (dim_b, n_hists) samples in the target domain, compute sinkhorn with multiple targets and fixed M if b is a matrix (return OT loss + dual variables in log) - M : ndarray, shape (dim_a, n_b) + M : ndarray, shape (dim_a, dim_b) loss matrix reg : float Regularization term >0 @@ -453,7 +456,7 @@ def _greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log Returns ------- - gamma : ndarray, shape (dim_a, n_b) + gamma : ndarray, shape (dim_a, dim_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -465,7 +468,7 @@ def _greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log >>> a=[.5, .5] >>> b=[.5, .5] >>> M=[[0., 1.], [1., 0.]] - >>> ot.bregman._greenkhorn(a, b, M, 1) + >>> ot.bregman.greenkhorn(a, b, M, 1) array([[0.36552929, 0.13447071], [0.13447071, 0.36552929]]) @@ -555,9 +558,9 @@ def _greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, log return G -def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, - warmstart=None, verbose=False, print_period=20, - log=False, **kwargs): +def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, + warmstart=None, verbose=False, print_period=20, + log=False, **kwargs): r""" Solve the entropic regularization OT problem with log stabilization @@ -573,7 +576,7 @@ def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, \gamma\geq 0 where : - - M is the (dim_a, n_b) metric cost matrix + - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -588,7 +591,7 @@ def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, samples weights in the source domain b : ndarray, shape (dim_b,) samples in the target domain - M : ndarray, shape (dim_a, n_b) + M : ndarray, shape (dim_a, dim_b) loss matrix reg : float Regularization term >0 @@ -607,7 +610,7 @@ def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, Returns ------- - gamma : ndarray, shape (dim_a, n_b) + gamma : ndarray, shape (dim_a, dim_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -619,7 +622,7 @@ def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, >>> a=[.5,.5] >>> b=[.5,.5] >>> M=[[0.,1.],[1.,0.]] - >>> ot.bregman._sinkhorn_stabilized(a, b, M, 1) + >>> ot.bregman.sinkhorn_stabilized(a, b, M, 1) array([[0.36552929, 0.13447071], [0.13447071, 0.36552929]]) @@ -658,8 +661,8 @@ def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, n_hists = 0 # init data - na = len(a) - nb = len(b) + dim_a = len(a) + dim_b = len(b) cpt = 0 if log: @@ -668,24 +671,25 @@ def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, # we assume that no distances are null except those of the diagonal of # distances if warmstart is None: - alpha, beta = np.zeros(na), np.zeros(nb) + alpha, beta = np.zeros(dim_a), np.zeros(dim_b) else: alpha, beta = warmstart if n_hists: - u, v = np.ones((na, n_hists)) / na, np.ones((nb, n_hists)) / nb + u = np.ones((dim_a, n_hists)) / dim_a + v = np.ones((dim_b, n_hists)) / dim_b else: - u, v = np.ones(na) / na, np.ones(nb) / nb + u, v = np.ones(dim_a) / dim_a, np.ones(dim_b) / dim_b def get_K(alpha, beta): """log space computation""" - return np.exp(-(M - alpha.reshape((na, 1)) - - beta.reshape((1, nb))) / reg) + return np.exp(-(M - alpha.reshape((dim_a, 1)) + - beta.reshape((1, dim_b))) / reg) def get_Gamma(alpha, beta, u, v): """log space gamma computation""" - return np.exp(-(M - alpha.reshape((na, 1)) - beta.reshape((1, nb))) - / reg + np.log(u.reshape((na, 1))) + np.log(v.reshape((1, nb)))) + return np.exp(-(M - alpha.reshape((dim_a, 1)) - beta.reshape((1, dim_b))) + / reg + np.log(u.reshape((dim_a, 1))) + np.log(v.reshape((1, dim_b)))) # print(np.min(K)) @@ -711,9 +715,9 @@ def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, else: alpha, beta = alpha + reg * np.log(u), beta + reg * np.log(v) if n_hists: - u, v = np.ones((na, n_hists)) / na, np.ones((nb, n_hists)) / nb + u, v = np.ones((dim_a, n_hists)) / dim_a, np.ones((dim_b, n_hists)) / dim_b else: - u, v = np.ones(na) / na, np.ones(nb) / nb + u, v = np.ones(dim_a) / dim_a, np.ones(dim_b) / dim_b K = get_K(alpha, beta) if cpt % print_period == 0: @@ -782,10 +786,10 @@ def _sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, return get_Gamma(alpha, beta, u, v) -def _sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, - numInnerItermax=100, tau=1e3, stopThr=1e-9, - warmstart=None, verbose=False, print_period=10, - log=False, **kwargs): +def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, + numInnerItermax=100, tau=1e3, stopThr=1e-9, + warmstart=None, verbose=False, print_period=10, + log=False, **kwargs): r""" Solve the entropic regularization optimal transport problem with log stabilization and epsilon scaling. @@ -802,7 +806,7 @@ def _sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, \gamma\geq 0 where : - - M is the (dim_a, n_b) metric cost matrix + - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - a and b are source and target weights (sum to 1) @@ -817,7 +821,7 @@ def _sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, samples weights in the source domain b : ndarray, shape (dim_b,) samples in the target domain - M : ndarray, shape (dim_a, n_b) + M : ndarray, shape (dim_a, dim_b) loss matrix reg : float Regularization term >0 @@ -840,7 +844,7 @@ def _sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, Returns ------- - gamma : ndarray, shape (dim_a, n_b) + gamma : ndarray, shape (dim_a, dim_b) Optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -852,7 +856,7 @@ def _sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, >>> a=[.5, .5] >>> b=[.5, .5] >>> M=[[0., 1.], [1., 0.]] - >>> ot.bregman._sinkhorn_epsilon_scaling(a, b, M, 1) + >>> ot.bregman.sinkhorn_epsilon_scaling(a, b, M, 1) array([[0.36552929, 0.13447071], [0.13447071, 0.36552929]]) @@ -915,8 +919,10 @@ def _sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, regi = get_reg(cpt) - G, logi = _sinkhorn_stabilized(a, b, M, regi, numItermax=numInnerItermax, stopThr=1e-9, warmstart=( - alpha, beta), verbose=False, print_period=20, tau=tau, log=True) + G, logi = sinkhorn_stabilized(a, b, M, regi, + numItermax=numInnerItermax, stopThr=1e-9, + warmstart=(alpha, beta), verbose=False, + print_period=20, tau=tau, log=True) alpha = logi['alpha'] beta = logi['beta'] @@ -1029,19 +1035,19 @@ def barycenter(A, M, reg, weights=None, method="sinkhorn", numItermax=10000, """ if method.lower() == 'sinkhorn': - return _barycenter(A, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, log=log, - **kwargs) + return barycenter_sinkhorn(A, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, log=log, + **kwargs) elif method.lower() == 'sinkhorn_stabilized': - return _barycenter_stabilized(A, M, reg, numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return barycenter_stabilized(A, M, reg, numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) else: raise ValueError("Unknown method '%s'." % method) -def _barycenter(A, M, reg, weights=None, numItermax=1000, - stopThr=1e-4, verbose=False, log=False): +def barycenter_sinkhorn(A, M, reg, weights=None, numItermax=1000, + stopThr=1e-4, verbose=False, log=False): r"""Compute the entropic regularized wasserstein barycenter of distributions A The function solves the following optimization problem: @@ -1134,8 +1140,8 @@ def _barycenter(A, M, reg, weights=None, numItermax=1000, return geometricBar(weights, UKv) -def _barycenter_stabilized(A, M, reg, tau=1e10, weights=None, numItermax=1000, - stopThr=1e-4, verbose=False, log=False): +def barycenter_stabilized(A, M, reg, tau=1e10, weights=None, numItermax=1000, + stopThr=1e-4, verbose=False, log=False): r"""Compute the entropic regularized wasserstein barycenter of distributions A with stabilization. @@ -1264,7 +1270,9 @@ def _barycenter_stabilized(A, M, reg, tau=1e10, weights=None, numItermax=1000, return q -def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1e-9, stabThr=1e-30, verbose=False, log=False): +def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, + stopThr=1e-9, stabThr=1e-30, verbose=False, + log=False): r"""Compute the entropic regularized wasserstein barycenter of distributions A where A is a collection of 2D images. @@ -1480,7 +1488,9 @@ def unmix(a, D, M, M0, h0, reg, reg0, alpha, numItermax=1000, return np.sum(K0, axis=1) -def empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numIterMax=10000, stopThr=1e-9, verbose=False, log=False, **kwargs): +def empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', + numIterMax=10000, stopThr=1e-9, verbose=False, + log=False, **kwargs): r''' Solve the entropic regularization optimal transport problem and return the OT matrix from empirical data @@ -1497,22 +1507,22 @@ def empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numI \gamma\geq 0 where : - - :math:`M` is the (dim_a, n_b) metric cost matrix + - :math:`M` is the (n_samples_a, n_samples_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - :math:`a` and :math:`b` are source and target weights (sum to 1) Parameters ---------- - X_s : ndarray, shape (dim_a, d) + X_s : ndarray, shape (n_samples_a, dim) samples in the source domain - X_t : ndarray, shape (dim_b, d) + X_t : ndarray, shape (n_samples_b, dim) samples in the target domain reg : float Regularization term >0 - a : ndarray, shape (dim_a,) + a : ndarray, shape (n_samples_a,) samples weights in the source domain - b : ndarray, shape (dim_b,) + b : ndarray, shape (n_samples_b,) samples weights in the target domain numItermax : int, optional Max number of iterations @@ -1526,7 +1536,7 @@ def empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numI Returns ------- - gamma : ndarray, shape (dim_a, n_b) + gamma : ndarray, shape (n_samples_a, n_samples_b) Regularized optimal transportation matrix for the given parameters log : dict log dictionary return only if log==True in parameters @@ -1587,7 +1597,7 @@ def empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', num \gamma\geq 0 where : - - :math:`M` is the (dim_a, n_b) metric cost matrix + - :math:`M` is the (n_samples_a, n_samples_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - :math:`a` and :math:`b` are source and target weights (sum to 1) @@ -1596,7 +1606,7 @@ def empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', num ---------- X_s : ndarray, shape (n_samples_a, dim) samples in the source domain - X_t : ndarray, shape (n_samples_b, d) + X_t : ndarray, shape (n_samples_b, dim) samples in the target domain reg : float Regularization term >0 @@ -1695,7 +1705,7 @@ def empirical_sinkhorn_divergence(X_s, X_t, reg, a=None, b=None, metric='sqeucli \gamma_b\geq 0 where : - - :math:`M` (resp. :math:`M_a, M_b`) is the (dim_a, n_b) metric cost matrix (resp (dim_a, ns) and (dim_b, nt)) + - :math:`M` (resp. :math:`M_a, M_b`) is the (n_samples_a, n_samples_b) metric cost matrix (resp (n_samples_a, n_samples_a) and (n_samples_b, n_samples_b)) - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - :math:`a` and :math:`b` are source and target weights (sum to 1) -- cgit v1.2.3 From 7efea812ad0b1c7e3783397dbd8f3ad802fb7ac2 Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Tue, 3 Sep 2019 17:26:30 +0200 Subject: same for unbalanced --- ot/unbalanced.py | 102 +++++++++++++++++++++++++++---------------------------- 1 file changed, 51 insertions(+), 51 deletions(-) diff --git a/ot/unbalanced.py b/ot/unbalanced.py index 3f71d28..25e4cf5 100644 --- a/ot/unbalanced.py +++ b/ot/unbalanced.py @@ -120,23 +120,23 @@ def sinkhorn_unbalanced(a, b, M, reg, reg_m, method='sinkhorn', numItermax=1000, """ if method.lower() == 'sinkhorn': - return _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) elif method.lower() == 'sinkhorn_stabilized': - return _sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, - numItermax=numItermax, - stopThr=stopThr, - verbose=verbose, - log=log, **kwargs) + return sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, + verbose=verbose, + log=log, **kwargs) elif method.lower() in ['sinkhorn_reg_scaling']: warnings.warn('Method not implemented yet. Using classic Sinkhorn Knopp') - return _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) else: raise ValueError("Unknown method '%s'." % method) @@ -241,29 +241,29 @@ def sinkhorn_unbalanced2(a, b, M, reg, reg_m, method='sinkhorn', if len(b.shape) < 2: b = b[:, None] if method.lower() == 'sinkhorn': - return _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) elif method.lower() == 'sinkhorn_stabilized': - return _sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, - numItermax=numItermax, - stopThr=stopThr, - verbose=verbose, - log=log, **kwargs) + return sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, + verbose=verbose, + log=log, **kwargs) elif method.lower() in ['sinkhorn_reg_scaling']: warnings.warn('Method not implemented yet. Using classic Sinkhorn Knopp') - return _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) else: raise ValueError('Unknown method %s.' % method) -def _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, numItermax=1000, - stopThr=1e-6, verbose=False, log=False, **kwargs): +def sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, numItermax=1000, + stopThr=1e-6, verbose=False, log=False, **kwargs): r""" Solve the entropic regularization unbalanced optimal transport problem and return the loss @@ -300,7 +300,7 @@ def _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, numItermax=1000, numItermax : int, optional Max number of iterations stopThr : float, optional - Stop threshol on error (>0) + Stop threshol on error (> 0) verbose : bool, optional Print information along iterations log : bool, optional @@ -439,9 +439,9 @@ def _sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, numItermax=1000, return u[:, None] * K * v[None, :] -def _sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, tau=1e5, numItermax=1000, - stopThr=1e-6, verbose=False, log=False, - **kwargs): +def sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, tau=1e5, numItermax=1000, + stopThr=1e-6, verbose=False, log=False, + **kwargs): r""" Solve the entropic regularization unbalanced optimal transport problem and return the loss @@ -653,9 +653,9 @@ def _sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, tau=1e5, numItermax=100 return ot_matrix -def _barycenter_unbalanced_stabilized(A, M, reg, reg_m, weights=None, tau=1e3, - numItermax=1000, stopThr=1e-6, - verbose=False, log=False): +def barycenter_unbalanced_stabilized(A, M, reg, reg_m, weights=None, tau=1e3, + numItermax=1000, stopThr=1e-6, + verbose=False, log=False): r"""Compute the entropic unbalanced wasserstein barycenter of A with stabilization. The function solves the following optimization problem: @@ -804,9 +804,9 @@ def _barycenter_unbalanced_stabilized(A, M, reg, reg_m, weights=None, tau=1e3, return q -def _barycenter_unbalanced(A, M, reg, reg_m, weights=None, - numItermax=1000, stopThr=1e-6, - verbose=False, log=False): +def barycenter_unbalanced_sinkhorn(A, M, reg, reg_m, weights=None, + numItermax=1000, stopThr=1e-6, + verbose=False, log=False): r"""Compute the entropic unbalanced wasserstein barycenter of A. The function solves the following optimization problem with a @@ -1001,22 +1001,22 @@ def barycenter_unbalanced(A, M, reg, reg_m, method="sinkhorn", weights=None, """ if method.lower() == 'sinkhorn': - return _barycenter_unbalanced(A, M, reg, reg_m, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return barycenter_unbalanced_sinkhorn(A, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) elif method.lower() == 'sinkhorn_stabilized': - return _barycenter_unbalanced_stabilized(A, M, reg, reg_m, - numItermax=numItermax, - stopThr=stopThr, - verbose=verbose, - log=log, **kwargs) + return barycenter_unbalanced_stabilized(A, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, + verbose=verbose, + log=log, **kwargs) elif method.lower() in ['sinkhorn_reg_scaling']: warnings.warn('Method not implemented yet. Using classic Sinkhorn Knopp') - return _barycenter_unbalanced(A, M, reg, reg_m, - numItermax=numItermax, - stopThr=stopThr, verbose=verbose, - log=log, **kwargs) + return barycenter_unbalanced(A, M, reg, reg_m, + numItermax=numItermax, + stopThr=stopThr, verbose=verbose, + log=log, **kwargs) else: raise ValueError("Unknown method '%s'." % method) -- cgit v1.2.3 From 49d9b5cf4eecefdc0fff4db6c43e85d16e478efb Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Tue, 3 Sep 2019 17:35:23 +0200 Subject: fix doctest examples --- ot/unbalanced.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/ot/unbalanced.py b/ot/unbalanced.py index 25e4cf5..d516dfc 100644 --- a/ot/unbalanced.py +++ b/ot/unbalanced.py @@ -326,7 +326,7 @@ def sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, numItermax=1000, >>> a=[.5, .5] >>> b=[.5, .5] >>> M=[[0., 1.],[1., 0.]] - >>> ot.unbalanced._sinkhorn_knopp_unbalanced(a, b, M, 1., 1.) + >>> ot.unbalanced.sinkhorn_knopp_unbalanced(a, b, M, 1., 1.) array([[0.51122823, 0.18807035], [0.18807035, 0.51122823]]) @@ -510,7 +510,7 @@ def sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, tau=1e5, numItermax=1000 >>> a=[.5, .5] >>> b=[.5, .5] >>> M=[[0., 1.],[1., 0.]] - >>> ot.unbalanced._sinkhorn_stabilized_unbalanced(a, b, M, 1., 1.) + >>> ot.unbalanced.sinkhorn_stabilized_unbalanced(a, b, M, 1., 1.) array([[0.51122823, 0.18807035], [0.18807035, 0.51122823]]) -- cgit v1.2.3 From e55232056a79de128583b87e65abc6d7a75fb298 Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Tue, 3 Sep 2019 18:15:25 +0200 Subject: add unbalanced_barycenter import --- ot/__init__.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ot/__init__.py b/ot/__init__.py index 7d9615a..99f288e 100644 --- a/ot/__init__.py +++ b/ot/__init__.py @@ -65,7 +65,7 @@ from . import unbalanced # OT functions from .lp import emd, emd2, emd_1d, emd2_1d, wasserstein_1d from .bregman import sinkhorn, sinkhorn2, barycenter -from .unbalanced import sinkhorn_unbalanced, barycenter_unbalanced +from .unbalanced import sinkhorn_unbalanced, barycenter_unbalanced, sinkhorn_unbalanced2 from .da import sinkhorn_lpl1_mm # utils functions -- cgit v1.2.3 From a96caaed079b6e368df090c3e3290398e8b4a99e Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Tue, 3 Sep 2019 18:15:38 +0200 Subject: update UOT paragraph in quickstart --- docs/source/quickstart.rst | 107 ++++++++++++++++++++++++++------------------- 1 file changed, 61 insertions(+), 46 deletions(-) diff --git a/docs/source/quickstart.rst b/docs/source/quickstart.rst index 1640d6a..e4ab3a3 100644 --- a/docs/source/quickstart.rst +++ b/docs/source/quickstart.rst @@ -2,13 +2,13 @@ Quick start guide ================= -In the following we provide some pointers about which functions and classes +In the following we provide some pointers about which functions and classes to use for different problems related to optimal transport (OT) and machine learning. We refer when we can to concrete examples in the documentation that are also available as notebooks on the POT Github. This document is not a tutorial on numerical optimal transport. For this we strongly -recommend to read the very nice book [15]_ . +recommend to read the very nice book [15]_ . Optimal transport and Wasserstein distance @@ -55,8 +55,8 @@ solver is quite efficient and uses sparsity of the solution. Examples of use for :any:`ot.emd` are available in : - :any:`auto_examples/plot_OT_2D_samples` - - :any:`auto_examples/plot_OT_1D` - - :any:`auto_examples/plot_OT_L1_vs_L2` + - :any:`auto_examples/plot_OT_1D` + - :any:`auto_examples/plot_OT_L1_vs_L2` Computing Wasserstein distance @@ -102,13 +102,13 @@ distance. An example of use for :any:`ot.emd2` is available in : - :any:`auto_examples/plot_compute_emd` - + Special cases ^^^^^^^^^^^^^ Note that the OT problem and the corresponding Wasserstein distance can in some -special cases be computed very efficiently. +special cases be computed very efficiently. For instance when the samples are in 1D, then the OT problem can be solved in :math:`O(n\log(n))` by using a simple sorting. In this case we provide the @@ -117,13 +117,13 @@ matrix and value. Note that since the solution is very sparse the :code:`sparse` parameter of :any:`ot.emd_1d` allows for solving and returning the solution for very large problems. Note that in order to compute directly the :math:`W_p` Wasserstein distance in 1D we provide the function :any:`ot.wasserstein_1d` that -takes :code:`p` as a parameter. +takes :code:`p` as a parameter. Another special case for estimating OT and Monge mapping is between Gaussian distributions. In this case there exists a close form solution given in Remark 2.29 in [15]_ and the Monge mapping is an affine function and can be also computed from the covariances and means of the source and target -distributions. In the case when the finite sample dataset is supposed gaussian, we provide +distributions. In the case when the finite sample dataset is supposed gaussian, we provide :any:`ot.da.OT_mapping_linear` that returns the parameters for the Monge mapping. @@ -176,7 +176,7 @@ solution of the resulting optimization problem can be expressed as: where :math:`u` and :math:`v` are vectors and :math:`K=\exp(-M/\lambda)` where the :math:`\exp` is taken component-wise. In order to solve the optimization problem, on can use an alternative projection algorithm called Sinkhorn-Knopp that can be very -efficient for large values if regularization. +efficient for large values if regularization. The Sinkhorn-Knopp algorithm is implemented in :any:`ot.sinkhorn` and :any:`ot.sinkhorn2` that return respectively the OT matrix and the value of the @@ -201,12 +201,12 @@ More details about the algorithms used are given in the following note. + :code:`method='sinkhorn'` calls :any:`ot.bregman.sinkhorn_knopp` the classic algorithm [2]_. + :code:`method='sinkhorn_stabilized'` calls :any:`ot.bregman.sinkhorn_stabilized` the - log stabilized version of the algorithm [9]_. + log stabilized version of the algorithm [9]_. + :code:`method='sinkhorn_epsilon_scaling'` calls :any:`ot.bregman.sinkhorn_epsilon_scaling` the epsilon scaling version - of the algorithm [9]_. + of the algorithm [9]_. + :code:`method='greenkhorn'` calls :any:`ot.bregman.greenkhorn` the - greedy sinkhorn verison of the algorithm [22]_. + greedy sinkhorn verison of the algorithm [22]_. In addition to all those variants of sinkhorn, we have another implementation solving the problem in the smooth dual or semi-dual in @@ -236,7 +236,7 @@ of algorithms in [18]_ [19]_. Examples of use for :any:`ot.sinkhorn` are available in : - :any:`auto_examples/plot_OT_2D_samples` - - :any:`auto_examples/plot_OT_1D` + - :any:`auto_examples/plot_OT_1D` - :any:`auto_examples/plot_OT_1D_smooth` - :any:`auto_examples/plot_stochastic` @@ -248,13 +248,13 @@ While entropic OT is the most common and favored in practice, there exist other kind of regularization. We provide in POT two specific solvers for other regularization terms, namely quadratic regularization and group lasso regularization. But we also provide in :any:`ot.optim` two generic solvers that allows solving any -smooth regularization in practice. +smooth regularization in practice. Quadratic regularization """""""""""""""""""""""" The first general regularization term we can solve is the quadratic -regularization of the form +regularization of the form .. math:: \Omega(\gamma)=\sum_{i,j} \gamma_{i,j}^2 @@ -264,7 +264,7 @@ densifying the OT matrix but it keeps some sort of sparsity that is lost with entropic regularization as soon as :math:`\lambda>0` [17]_. This problem can be solved with POT using solvers from :any:`ot.smooth`, more specifically functions :any:`ot.smooth.smooth_ot_dual` or -:any:`ot.smooth.smooth_ot_semi_dual` with parameter :code:`reg_type='l2'` to +:any:`ot.smooth.smooth_ot_semi_dual` with parameter :code:`reg_type='l2'` to choose the quadratic regularization. .. hint:: @@ -300,7 +300,7 @@ gradient algorithm [7]_ in function .. hint:: Examples of group Lasso regularization are available in : - - :any:`auto_examples/plot_otda_classes` + - :any:`auto_examples/plot_otda_classes` - :any:`auto_examples/plot_otda_d2` @@ -311,7 +311,7 @@ Finally we propose in POT generic solvers that can be used to solve any regularization as long as you can provide a function computing the regularization and a function computing its gradient (or sub-gradient). -In order to solve +In order to solve .. math:: \gamma^* = arg\min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + \lambda\Omega(\gamma) @@ -336,12 +336,12 @@ Another generic solver is proposed to solve the problem where :math:`\Omega_e` is the entropic regularization. In this case we use a generalized conditional gradient [7]_ implemented in :any:`ot.optim.gcg` that does not linearize the entropic term but -relies on :any:`ot.sinkhorn` for its iterations. +relies on :any:`ot.sinkhorn` for its iterations. .. hint:: An example of generic solvers are available in : - - :any:`auto_examples/plot_optim_OTreg` + - :any:`auto_examples/plot_optim_OTreg` Wasserstein Barycenters @@ -382,7 +382,7 @@ solver :any:`ot.lp.barycenter` that rely on generic LP solvers. By default the function uses :any:`scipy.optimize.linprog`, but more efficient LP solvers from cvxopt can be also used by changing parameter :code:`solver`. Note that this problem requires to solve a very large linear program and can be very slow in -practice. +practice. Similarly to the OT problem, OT barycenters can be computed in the regularized case. When using entropic regularization is used, the problem can be solved with a @@ -403,11 +403,11 @@ operators. We provide an implementation of this algorithm in function Examples of Wasserstein (:any:`ot.lp.barycenter`) and regularized Wasserstein barycenter (:any:`ot.bregman.barycenter`) computation are available in : - - :any:`auto_examples/plot_barycenter_1D` - - :any:`auto_examples/plot_barycenter_lp_vs_entropic` + - :any:`auto_examples/plot_barycenter_1D` + - :any:`auto_examples/plot_barycenter_lp_vs_entropic` An example of convolutional barycenter - (:any:`ot.bregman.convolutional_barycenter2d`) computation + (:any:`ot.bregman.convolutional_barycenter2d`) computation for 2D images is available in : @@ -451,13 +451,13 @@ optimal mapping is still an open problem in the general case but has been proven for smooth distributions by Brenier in his eponym `theorem `__. We provide in :any:`ot.da` several solvers for smooth Monge mapping estimation and domain -adaptation from discrete distributions. +adaptation from discrete distributions. Monge Mapping estimation ^^^^^^^^^^^^^^^^^^^^^^^^ We now discuss several approaches that are implemented in POT to estimate or -approximate a Monge mapping from finite distributions. +approximate a Monge mapping from finite distributions. First note that when the source and target distributions are supposed to be Gaussian distributions, there exists a close form solution for the mapping and its an @@ -513,16 +513,16 @@ A list of the provided implementation is given in the following note. Here is a list of the OT mapping classes inheriting from :any:`ot.da.BaseTransport` - + * :any:`ot.da.EMDTransport` : Barycentric mapping with EMD transport * :any:`ot.da.SinkhornTransport` : Barycentric mapping with Sinkhorn transport * :any:`ot.da.SinkhornL1l2Transport` : Barycentric mapping with Sinkhorn + group Lasso regularization [5]_ * :any:`ot.da.SinkhornLpl1Transport` : Barycentric mapping with Sinkhorn + - non convex group Lasso regularization [5]_ + non convex group Lasso regularization [5]_ * :any:`ot.da.LinearTransport` : Linear mapping estimation between Gaussians [14]_ - * :any:`ot.da.MappingTransport` : Nonlinear mapping estimation [8]_ + * :any:`ot.da.MappingTransport` : Nonlinear mapping estimation [8]_ .. hint:: @@ -550,7 +550,7 @@ consist in finding a linear projector optimizing the following criterion .. math:: P = \text{arg}\min_P \frac{\sum_i OT_e(\mu_i\#P,\mu_i\#P)}{\sum_{i,j\neq i} OT_e(\mu_i\#P,\mu_j\#P)} - + where :math:`\#` is the push-forward operator, :math:`OT_e` is the entropic OT loss and :math:`\mu_i` is the distribution of samples from class :math:`i`. :math:`P` is also constrained to @@ -575,10 +575,10 @@ respectively. Note that we also provide the Fisher discriminant estimator in Unbalanced optimal transport ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -Unbalanced OT is a relaxation of the original OT problem where the violation of +Unbalanced OT is a relaxation of the entropy regularized OT problem where the violation of the constraint on the marginals is added to the objective of the optimization problem: - + .. math:: \min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + reg\cdot\Omega(\gamma) + \alpha KL(\gamma 1, a) + \alpha KL(\gamma^T 1, b) @@ -589,9 +589,24 @@ where KL is the Kullback-Leibler divergence. This formulation allows for computing approximate mapping between distributions that do not have the same amount of mass. Interestingly the problem can be solved with a generalization of the Bregman projections algorithm [10]_. We provide a solver for unbalanced OT -in :any:`ot.unbalanced` and more specifically -in function :any:`ot.sinkhorn_unbalanced`. A solver for unbalanced OT barycenter -is available in :any:`ot.barycenter_unbalanced`. +in :any:`ot.unbalanced`. Computing the optimal transport +plan or the transport cost is similar to the balanced case. The Sinkhorn-Knopp +algorithm is implemented in :any:`ot.sinkhorn_unbalanced` and :any:`ot.sinkhorn_unbalanced2` +that return respectively the OT matrix and the value of the +linear term. Note that the regularization parameter :math:`\alpha` in the +equation above is given to those functions with the parameter :code:`reg_m`. + +Similarly, Unbalanced OT barycenters can be computed using :any:`ot.barycenter_unbalanced`. + +.. note:: + The main function to solve entropic regularized OT is :any:`ot.sinkhorn_unbalanced`. + This function is a wrapper and the parameter :code:`method` help you select + the actual algorithm used to solve the problem: + + + :code:`method='sinkhorn'` calls :any:`ot.unbalanced.sinkhorn_knopp_unbalanced` + the generalized Sinkhorn algorithm [10]_. + + :code:`method='sinkhorn_stabilized'` calls :any:`ot.unbalanced.sinkhorn_stabilized_unbalanced` + the log stabilized version of the algorithm [10]_. .. hint:: @@ -636,7 +651,7 @@ barycenters that can be expressed as where :math:`Ck` is the distance matrix between samples in distribution :math:`k`. Note that interestingly the barycenter is defined as a symmetric -positive matrix. We provide a block coordinate optimization procedure in +positive matrix. We provide a block coordinate optimization procedure in :any:`ot.gromov.gromov_barycenters` and :any:`ot.gromov.entropic_gromov_barycenters` for non-regularized and regularized barycenters respectively. @@ -654,19 +669,19 @@ The implementations of FGW and FGW barycenter is provided in functions Examples of computation of GW, regularized G and FGW are available in : - :any:`auto_examples/plot_gromov` - - :any:`auto_examples/plot_fgw` + - :any:`auto_examples/plot_fgw` Examples of GW, regularized GW and FGW barycenters are available in : - :any:`auto_examples/plot_gromov_barycenter` - - :any:`auto_examples/plot_barycenter_fgw` + - :any:`auto_examples/plot_barycenter_fgw` GPU acceleration ^^^^^^^^^^^^^^^^ We provide several implementation of our OT solvers in :any:`ot.gpu`. Those -implementations use the :code:`cupy` toolbox that obviously need to be installed. +implementations use the :code:`cupy` toolbox that obviously need to be installed. .. note:: @@ -701,7 +716,7 @@ FAQ 1. **How to solve a discrete optimal transport problem ?** The solver for discrete OT is the function :py:mod:`ot.emd` that returns - the OT transport matrix. If you want to solve a regularized OT you can + the OT transport matrix. If you want to solve a regularized OT you can use :py:mod:`ot.sinkhorn`. @@ -714,9 +729,9 @@ FAQ T=ot.emd(a,b,M) # exact linear program T_reg=ot.sinkhorn(a,b,M,reg) # entropic regularized OT - More detailed examples can be seen on this example: + More detailed examples can be seen on this example: :doc:`auto_examples/plot_OT_2D_samples` - + 2. **pip install POT fails with error : ImportError: No module named Cython.Build** @@ -726,7 +741,7 @@ FAQ installing POT. Note that this problem do not occur when using conda-forge since the packages - there are pre-compiled. + there are pre-compiled. See `Issue #59 `__ for more details. @@ -751,7 +766,7 @@ FAQ In order to limit import time and hard dependencies in POT. we do not import some sub-modules automatically with :code:`import ot`. In order to use the acceleration in :any:`ot.gpu` you need first to import is with - :code:`import ot.gpu`. + :code:`import ot.gpu`. See `Issue #85 `__ and :any:`ot.gpu` for more details. @@ -763,7 +778,7 @@ References .. [1] Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W. (2011, December). `Displacement nterpolation using Lagrangian mass transport `__. - In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM. + In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM. .. [2] Cuturi, M. (2013). `Sinkhorn distances: Lightspeed computation of optimal transport `__. In Advances @@ -874,4 +889,4 @@ References .. [24] Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N. (2019). `Optimal Transport for structured data with application on graphs `__ Proceedings - of the 36th International Conference on Machine Learning (ICML). \ No newline at end of file + of the 36th International Conference on Machine Learning (ICML). -- cgit v1.2.3 From b639e3eba2e88e20b5d2df417368200e4fc5157c Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Tue, 3 Sep 2019 18:39:03 +0200 Subject: update UOT text + add frogner ref --- docs/source/quickstart.rst | 49 ++++++++++++++++++++++++++++++++++++++-------- 1 file changed, 41 insertions(+), 8 deletions(-) diff --git a/docs/source/quickstart.rst b/docs/source/quickstart.rst index e4ab3a3..9729664 100644 --- a/docs/source/quickstart.rst +++ b/docs/source/quickstart.rst @@ -577,10 +577,10 @@ Unbalanced optimal transport Unbalanced OT is a relaxation of the entropy regularized OT problem where the violation of the constraint on the marginals is added to the objective of the optimization -problem: +problem. The unbalanced OT metric between two histograms a and b is defined as [25]_ [10]_: .. math:: - \min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + reg\cdot\Omega(\gamma) + \alpha KL(\gamma 1, a) + \alpha KL(\gamma^T 1, b) + W_u(a, b) = \min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + reg\cdot\Omega(\gamma) + \alpha KL(\gamma 1, a) + \alpha KL(\gamma^T 1, b) s.t. \quad \gamma\geq 0 @@ -596,26 +596,55 @@ that return respectively the OT matrix and the value of the linear term. Note that the regularization parameter :math:`\alpha` in the equation above is given to those functions with the parameter :code:`reg_m`. -Similarly, Unbalanced OT barycenters can be computed using :any:`ot.barycenter_unbalanced`. .. note:: - The main function to solve entropic regularized OT is :any:`ot.sinkhorn_unbalanced`. + The main function to solve entropic regularized UOT is :any:`ot.sinkhorn_unbalanced`. This function is a wrapper and the parameter :code:`method` help you select the actual algorithm used to solve the problem: + :code:`method='sinkhorn'` calls :any:`ot.unbalanced.sinkhorn_knopp_unbalanced` - the generalized Sinkhorn algorithm [10]_. + the generalized Sinkhorn algorithm [25]_ [10]_. + :code:`method='sinkhorn_stabilized'` calls :any:`ot.unbalanced.sinkhorn_stabilized_unbalanced` the log stabilized version of the algorithm [10]_. .. hint:: - Examples of the use of :any:`ot.sinkhorn_unbalanced` and - :any:`ot.barycenter_unbalanced` are available in : + Examples of the use of :any:`ot.sinkhorn_unbalanced` are available in : - :any:`auto_examples/plot_UOT_1D` - - :any:`auto_examples/plot_UOT_barycenter_1D` + + +Unbalanced Barycenters +^^^^^^^^^^^^^^^^^^^^^^ + +As with balanced distributions, we can define a barycenter of a set of +histograms with different masses as a Fréchet Mean: + + .. math:: + \min_{\mu} \quad \sum_{k} w_kW_u(\mu,\mu_k) + +Where :math:`W_u` is the unbalanced Wasserstein metric defined above. This problem +can also be solved using generalized version of Sinkhorn's algorithm and it is +implemented the main function :any:`ot.barycenter_unbalanced`. + + +.. note:: + The main function to compute UOT barycenters is :any:`ot.barycenter_unbalanced`. + This function is a wrapper and the parameter :code:`method` help you select + the actual algorithm used to solve the problem: + + + :code:`method='sinkhorn'` calls :any:`ot.unbalanced.barycenter_unbalanced_sinkhorn_unbalanced` + the generalized Sinkhorn algorithm [10]_. + + :code:`method='sinkhorn_stabilized'` calls :any:`ot.unbalanced.barycenter_unbalanced_stabilized` + the log stabilized version of the algorithm [10]_. + + +.. hint:: + + Examples of the use of :any:`ot.barycenter_unbalanced` are available in : + + - :any:`auto_examples/plot_UOT_barycenter_1D` Gromov-Wasserstein @@ -890,3 +919,7 @@ References (2019). `Optimal Transport for structured data with application on graphs `__ Proceedings of the 36th International Conference on Machine Learning (ICML). + +.. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : + Learning with a Wasserstein Loss, Advances in Neural Information + Processing Systems (NIPS) 2015 -- cgit v1.2.3 From e0c935a865a57bc4603144b27f1b58cbfba87760 Mon Sep 17 00:00:00 2001 From: Hicham Janati Date: Wed, 4 Sep 2019 10:28:04 +0200 Subject: improve doc --- docs/source/quickstart.rst | 10 ++++------ ot/__init__.py | 3 ++- ot/bregman.py | 40 ++++++++++++++++++++++------------------ 3 files changed, 28 insertions(+), 25 deletions(-) diff --git a/docs/source/quickstart.rst b/docs/source/quickstart.rst index 9729664..978eaff 100644 --- a/docs/source/quickstart.rst +++ b/docs/source/quickstart.rst @@ -577,10 +577,10 @@ Unbalanced optimal transport Unbalanced OT is a relaxation of the entropy regularized OT problem where the violation of the constraint on the marginals is added to the objective of the optimization -problem. The unbalanced OT metric between two histograms a and b is defined as [25]_ [10]_: +problem. The unbalanced OT metric between two unbalanced histograms a and b is defined as [25]_ [10]_: .. math:: - W_u(a, b) = \min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + reg\cdot\Omega(\gamma) + \alpha KL(\gamma 1, a) + \alpha KL(\gamma^T 1, b) + W_u(a, b) = \min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + reg\cdot\Omega(\gamma) + reg_m KL(\gamma 1, a) + reg_m KL(\gamma^T 1, b) s.t. \quad \gamma\geq 0 @@ -593,13 +593,11 @@ in :any:`ot.unbalanced`. Computing the optimal transport plan or the transport cost is similar to the balanced case. The Sinkhorn-Knopp algorithm is implemented in :any:`ot.sinkhorn_unbalanced` and :any:`ot.sinkhorn_unbalanced2` that return respectively the OT matrix and the value of the -linear term. Note that the regularization parameter :math:`\alpha` in the -equation above is given to those functions with the parameter :code:`reg_m`. - +linear term. .. note:: The main function to solve entropic regularized UOT is :any:`ot.sinkhorn_unbalanced`. - This function is a wrapper and the parameter :code:`method` help you select + This function is a wrapper and the parameter :code:`method` helps you select the actual algorithm used to solve the problem: + :code:`method='sinkhorn'` calls :any:`ot.unbalanced.sinkhorn_knopp_unbalanced` diff --git a/ot/__init__.py b/ot/__init__.py index 99f288e..df0ef8a 100644 --- a/ot/__init__.py +++ b/ot/__init__.py @@ -77,4 +77,5 @@ __all__ = ["emd", "emd2", 'emd_1d', "sinkhorn", "sinkhorn2", "utils", 'datasets' 'bregman', 'lp', 'tic', 'toc', 'toq', 'gromov', 'emd_1d', 'emd2_1d', 'wasserstein_1d', 'dist', 'unif', 'barycenter', 'sinkhorn_lpl1_mm', 'da', 'optim', - 'sinkhorn_unbalanced', "barycenter_unbalanced"] + 'sinkhorn_unbalanced', 'barycenter_unbalanced', + 'sinkhorn_unbalanced2'] diff --git a/ot/bregman.py b/ot/bregman.py index 02aeb6d..2cd832b 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -35,7 +35,7 @@ def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - a and b are source and target weights (sum to 1) + - a and b are source and target weights (histograms, both sum to 1) The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [2]_ @@ -143,7 +143,7 @@ def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000, - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - a and b are source and target weights (sum to 1) + - a and b are source and target weights (histograms, both sum to 1) The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [2]_ @@ -251,7 +251,7 @@ def sinkhorn_knopp(a, b, M, reg, numItermax=1000, - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - a and b are source and target weights (sum to 1) + - a and b are source and target weights (histograms, both sum to 1) The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [2]_ @@ -432,7 +432,7 @@ def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False, - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - a and b are source and target weights (sum to 1) + - a and b are source and target weights (histograms, both sum to 1) @@ -578,7 +578,8 @@ def sinkhorn_stabilized(a, b, M, reg, numItermax=1000, tau=1e3, stopThr=1e-9, - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - a and b are source and target weights (sum to 1) + - a and b are source and target weights (histograms, both sum to 1) + The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [2]_ but with the log stabilization @@ -808,7 +809,8 @@ def sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=100, epsilon0=1e4, - M is the (dim_a, dim_b) metric cost matrix - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - a and b are source and target weights (sum to 1) + - a and b are source and target weights (histograms, both sum to 1) + The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [2]_ but with the log stabilization @@ -1229,7 +1231,6 @@ def barycenter_stabilized(A, M, reg, tau=1e10, weights=None, numItermax=1000, absorbing = False if (u > tau).any() or (v > tau).any(): absorbing = True - print("YEAH absorbing") alpha = alpha + reg * np.log(np.max(u, 1)) beta = beta + reg * np.log(np.max(v, 1)) K = np.exp((alpha[:, None] + beta[None, :] - @@ -1394,9 +1395,12 @@ def unmix(a, D, M, M0, h0, reg, reg0, alpha, numItermax=1000, where : - :math:`W_{M,reg}(\cdot,\cdot)` is the entropic regularized Wasserstein distance with M loss matrix (see ot.bregman.sinkhorn) - - :math:`\mathbf{a}` is an observed distribution, :math:`\mathbf{h}_0` is aprior on unmixing - - reg and :math:`\mathbf{M}` are respectively the regularization term and the cost matrix for OT data fitting - - reg0 and :math:`\mathbf{M0}` are respectively the regularization term and the cost matrix for regularization + - :math: `\mathbf{D}` is a dictionary of `n_atoms` atoms of dimension `dim_a`, its expected shape is `(dim_a, n_atoms)` + - :math:`\mathbf{h}` is the estimated unmixing of dimension `n_atoms` + - :math:`\mathbf{a}` is an observed distribution of dimension `dim_a` + - :math:`\mathbf{h}_0` is a prior on `h` of dimension `dim_prior` + - reg and :math:`\mathbf{M}` are respectively the regularization term and the cost matrix (dim_a, dim_a) for OT data fitting + - reg0 and :math:`\mathbf{M0}` are respectively the regularization term and the cost matrix (dim_prior, n_atoms) regularization - :math:`\\alpha`weight data fitting and regularization The optimization problem is solved suing the algorithm described in [4] @@ -1404,16 +1408,16 @@ def unmix(a, D, M, M0, h0, reg, reg0, alpha, numItermax=1000, Parameters ---------- - a : ndarray, shape (n_observed) - observed distribution - D : ndarray, shape (dim, dim) + a : ndarray, shape (dim_a) + observed distribution (histogram, sums to 1) + D : ndarray, shape (dim_a, n_atoms) dictionary matrix - M : ndarray, shape (dim, dim) + M : ndarray, shape (dim_a, dim_a) loss matrix - M0 : ndarray, shape (n_observed, n_observed) + M0 : ndarray, shape (n_atoms, dim_prior) loss matrix - h0 : ndarray, shape (dim,) - prior on h + h0 : ndarray, shape (n_atoms,) + prior on the estimated unmixing h reg : float Regularization term >0 (Wasserstein data fitting) reg0 : float @@ -1432,7 +1436,7 @@ def unmix(a, D, M, M0, h0, reg, reg0, alpha, numItermax=1000, Returns ------- - a : ndarray, shape (dim,) + h : ndarray, shape (n_atoms,) Wasserstein barycenter log : dict log dictionary return only if log==True in parameters -- cgit v1.2.3