add empirical sinkhorn and sinkhorn divergence functions

README.md

@@ -15,7 +15,8 @@ This open source Python library provide several solvers for optimization problem
 It provides the following solvers:
 
 * OT Network Flow solver for the linear program/ Earth Movers Distance [1].
-* Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] and stabilized version [9][10] and greedy SInkhorn [22] with optional GPU implementation (requires cupy).
+* Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2], stabilized version [9][10] and greedy Sinkhorn [22] with optional GPU implementation (requires cupy).
+* Sinkhorn divergence [23] and entropic regularization OT  from empirical data.
 * Smooth optimal transport solvers (dual and semi-dual) for KL and squared L2 regularizations [17].
 * Non regularized Wasserstein barycenters [16] with LP solver (only small scale).
 * Bregman projections for Wasserstein barycenter [3], convolutional barycenter [21] and unmixing [4].
@@ -230,3 +231,5 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 [21] Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A. & Guibas, L. (2015). [Convolutional wasserstein distances: Efficient optimal transportation on geometric domains](https://dl.acm.org/citation.cfm?id=2766963). ACM Transactions on Graphics (TOG), 34(4), 66.
 
 [22] J. Altschuler, J.Weed, P. Rigollet, (2017) [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://papers.nips.cc/paper/6792-near-linear-time-approximation-algorithms-for-optimal-transport-via-sinkhorn-iteration.pdf), Advances in Neural Information Processing Systems (NIPS) 31
+
+[23] Aude, G., Peyré, G., Cuturi, M., [Learning Generative Models with Sinkhorn Divergences](https://arxiv.org/abs/1706.00292), Proceedings of the Twenty-First International Conference on Artficial Intelligence and Statistics, (AISTATS) 21, 2018

examples/plot_OT_2D_samples.py

@@ -10,6 +10,7 @@
 """
 
 # Author: Remi Flamary <remi.flamary@unice.fr>
+#         Kilian Fatras <kilian.fatras@irisa.fr>
 #
 # License: MIT License
 
@@ -100,3 +101,28 @@
 pl.title('OT matrix Sinkhorn with samples')
 
 pl.show()
+
+
+##############################################################################
+# Emprirical Sinkhorn
+# ----------------
+
+#%% sinkhorn
+
+# reg term
+lambd = 1e-3
+
+Ges = ot.bregman.empirical_sinkhorn(xs, xt, lambd)
+
+pl.figure(7)
+pl.imshow(Ges, interpolation='nearest')
+pl.title('OT matrix empirical sinkhorn')
+
+pl.figure(8)
+ot.plot.plot2D_samples_mat(xs, xt, Ges, color=[.5, .5, 1])
+pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+pl.legend(loc=0)
+pl.title('OT matrix Sinkhorn from samples')
+
+pl.show()

ot/bregman.py

@@ -5,10 +5,12 @@
 
 # Author: Remi Flamary <remi.flamary@unice.fr>
 #         Nicolas Courty <ncourty@irisa.fr>
+#         Kilian Fatras <kilian.fatras@irisa.fr>
 #
 # License: MIT License
 
 import numpy as np
+from .utils import unif, dist
 
 
 def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000,
@@ -1296,3 +1298,302 @@ def unmix(a, D, M, M0, h0, reg, reg0, alpha, numItermax=1000,
         return np.sum(K0, axis=1), log
     else:
         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):
+    '''
+    Solve the entropic regularization optimal transport problem and return the
+    OT matrix from empirical data
+
+    The function solves the following optimization problem:
+
+    .. math::
+        \gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)
+
+        s.t. \gamma 1 = a
+
+             \gamma^T 1= b
+
+             \gamma\geq 0
+    where :
+
+    - :math:`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})`
+    - :math:`a` and :math:`b` are source and target weights (sum to 1)
+
+
+    Parameters
+    ----------
+    X_s : np.ndarray (ns, d)
+        samples in the source domain
+    X_t : np.ndarray (nt, d)
+        samples in the target domain
+    reg : float
+        Regularization term >0
+    a : np.ndarray (ns,)
+        samples weights in the source domain
+    b : np.ndarray (nt,)
+        samples weights in the target domain
+    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
+    -------
+    gamma : (ns x nt) ndarray
+        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 = 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))
+    >>> emp_sinkhorn = empirical_sinkhorn(X_s, X_t, reg, verbose=False)
+    >>> print(emp_sinkhorn)
+    >>> [[4.99977301e-01 2.26989344e-05]
+        [2.26989344e-05 4.99977301e-01]]
+
+
+    References
+    ----------
+
+    .. [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.
+
+    .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.
+    '''
+
+    if a is None:
+        a = unif(np.shape(X_s)[0])
+    if b is None:
+        b = unif(np.shape(X_t)[0])
+
+    M = dist(X_s, X_t, metric=metric)
+
+    if log:
+        pi, log = sinkhorn(a, b, M, reg, numItermax=numIterMax, stopThr=stopThr, verbose=verbose, log=True, **kwargs)
+        return pi, log
+    else:
+        pi = sinkhorn(a, b, M, reg, numItermax=numIterMax, stopThr=stopThr, verbose=verbose, log=False, **kwargs)
+        return pi
+
+
+def empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numIterMax=10000, stopThr=1e-9, verbose=False, log=False, **kwargs):
+    '''
+    Solve the entropic regularization optimal transport problem from empirical
+    data and return the OT loss
+
+
+    The function solves the following optimization problem:
+
+    .. math::
+        W = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)
+
+        s.t. \gamma 1 = a
+
+             \gamma^T 1= b
+
+             \gamma\geq 0
+    where :
+
+    - :math:`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})`
+    - :math:`a` and :math:`b` are source and target weights (sum to 1)
+
+
+    Parameters
+    ----------
+    X_s : np.ndarray (ns, d)
+        samples in the source domain
+    X_t : np.ndarray (nt, d)
+        samples in the target domain
+    reg : float
+        Regularization term >0
+    a : np.ndarray (ns,)
+        samples weights in the source domain
+    b : np.ndarray (nt,)
+        samples weights in the target domain
+    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
+    -------
+    gamma : (ns x nt) ndarray
+        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 = 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))
+    >>> loss_sinkhorn = empirical_sinkhorn2(X_s, X_t, reg, verbose=False)
+    >>> print(loss_sinkhorn)
+    >>> [4.53978687e-05]
+
+
+    References
+    ----------
+
+    .. [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.
+
+    .. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.
+    '''
+
+    if a is None:
+        a = unif(np.shape(X_s)[0])
+    if b is None:
+        b = unif(np.shape(X_t)[0])
+
+    M = dist(X_s, X_t, metric=metric)
+
+    if log:
+        sinkhorn_loss, log = sinkhorn2(a, b, M, reg, numItermax=numIterMax, stopThr=stopThr, verbose=verbose, log=log, **kwargs)
+        return sinkhorn_loss, log
+    else:
+        sinkhorn_loss = sinkhorn2(a, b, M, reg, numItermax=numIterMax, stopThr=stopThr, verbose=verbose, log=log, **kwargs)
+        return sinkhorn_loss
+
+
+def empirical_sinkhorn_divergence(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numIterMax=10000, stopThr=1e-9, verbose=False, log=False, **kwargs):
+    '''
+    Compute the sinkhorn divergence loss from empirical data
+
+    The function solves the following optimization problems and return the
+    sinkhorn divergence :math:`S`:
+
+    .. math::
+
+        W &= \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)
+
+        W_a &= \min_{\gamma_a} <\gamma_a,M_a>_F + reg\cdot\Omega(\gamma_a)
+
+        W_b &= \min_{\gamma_b} <\gamma_b,M_b>_F + reg\cdot\Omega(\gamma_b)
+
+        S &= W - 1/2 * (W_a + W_b)
+
+    .. math::
+        s.t. \gamma 1 = a
+
+             \gamma^T 1= b
+
+             \gamma\geq 0
+
+             \gamma_a 1 = a
+
+             \gamma_a^T 1= a
+
+             \gamma_a\geq 0
+
+             \gamma_b 1 = b
+
+             \gamma_b^T 1= b
+
+             \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:`\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 : np.ndarray (ns, d)
+        samples in the source domain
+    X_t : np.ndarray (nt, d)
+        samples in the target domain
+    reg : float
+        Regularization term >0
+    a : np.ndarray (ns,)
+        samples weights in the source domain
+    b : np.ndarray (nt,)
+        samples weights in the target domain
+    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
+    -------
+    gamma : (ns x nt) ndarray
+        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
+    >>> 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))
+    >>> emp_sinkhorn_div = empirical_sinkhorn_divergence(X_s, X_t, reg)
+    >>> print(emp_sinkhorn_div)
+    >>> [2.99977435]
+
+
+    References
+    ----------
+
+    .. [23] Aude Genevay, Gabriel Peyré, Marco Cuturi, Learning Generative Models with Sinkhorn Divergences,  Proceedings of the Twenty-First International Conference on Artficial Intelligence and Statistics, (AISTATS) 21, 2018
+    '''
+    if log:
+        sinkhorn_loss_ab, log_ab = empirical_sinkhorn2(X_s, X_t, reg, a, b, metric=metric, numIterMax=numIterMax, stopThr=1e-9, verbose=verbose, log=log, **kwargs)
+
+        sinkhorn_loss_a, log_a = empirical_sinkhorn2(X_s, X_s, reg, a, b, metric=metric, numIterMax=numIterMax, stopThr=1e-9, verbose=verbose, log=log, **kwargs)
+
+        sinkhorn_loss_b, log_b = empirical_sinkhorn2(X_t, X_t, reg, a, b, metric=metric, numIterMax=numIterMax, stopThr=1e-9, verbose=verbose, log=log, **kwargs)
+
+        sinkhorn_div = sinkhorn_loss_ab - 1 / 2 * (sinkhorn_loss_a + sinkhorn_loss_b)
+
+        log = {}
+        log['sinkhorn_loss_ab'] = sinkhorn_loss_ab
+        log['sinkhorn_loss_a'] = sinkhorn_loss_a
+        log['sinkhorn_loss_b'] = sinkhorn_loss_b
+        log['log_sinkhorn_ab'] = log_ab
+        log['log_sinkhorn_a'] = log_a
+        log['log_sinkhorn_b'] = log_b
+
+        return max(0, sinkhorn_div), log
+
+    else:
+        sinkhorn_loss_ab = empirical_sinkhorn2(X_s, X_t, reg, a, b, metric=metric, numIterMax=numIterMax, stopThr=1e-9, verbose=verbose, log=log, **kwargs)
+
+        sinkhorn_loss_a = empirical_sinkhorn2(X_s, X_s, reg, a, b, metric=metric, numIterMax=numIterMax, stopThr=1e-9, verbose=verbose, log=log, **kwargs)
+
+        sinkhorn_loss_b = empirical_sinkhorn2(X_t, X_t, reg, a, b, metric=metric, numIterMax=numIterMax, stopThr=1e-9, verbose=verbose, log=log, **kwargs)
+
+        sinkhorn_div = sinkhorn_loss_ab - 1 / 2 * (sinkhorn_loss_a + sinkhorn_loss_b)
+        return max(0, sinkhorn_div)

ot/stochastic.py

@@ -348,8 +348,11 @@ def solve_semi_dual_entropic(a, b, M, reg, method, numItermax=10000, lr=None,
 
     .. math::
         \gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)
+
         s.t. \gamma 1 = a
+
              \gamma^T 1= b
+
              \gamma \geq 0
 
     Where :