In this paper, a regularization of Wasserstein barycenters for random measures supported on @math is introduced via convex penalization. The existence and uniqueness of such barycenters is first proved for a large class of penalization functions. The Bregman divergence associated to the penalization term is then considered to obtain a stability result on penalized barycenters. This allows the comparison of data made of @math absolutely continuous probability measures, within the more realistic setting where one only has access to a dataset of random variables sampled from unknown distributions. The convergence of the penalized empirical barycenter of a set of @math iid random probability measures towards its population counterpart is finally analyzed. This approach is shown to be appropriate for the statistical analysis of either discrete or absolutely continuous random measures. It also allows to construct, from a set of discrete measures, consistent estimators of population Wasserstein barycenters that are absolutely continuous.
* Penalization of the transport map. Alternatively, regularized barycenters may be obtained by adding a convex regularization on optimal transport plans (that is on @math in ) when computing the Wasserstein distance between probability measures. This approach leads to the notion of regularized transportation problems and Wasserstein costs. It has recently gained popularity in the literature of image processing and machine learning and has been considered to compute smoothed Wasserstein barycenters of discrete measures @cite_19 @cite_21 . Such penalizations acting on transport plans nevertheless have an indirect influence on the regularity of the Wasserstein barycenter.
@cite_19 @cite_21
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This article introduces a generalization of discrete Optimal Transport that includes a regularity penalty and a relaxation of the bijectivity constraint. The corresponding transport plan is solved by minimizing an energy which is a convexification of an integer optimization problem. We propose to use a proximal splitting scheme to perform the minimization on large scale imaging problems. For un-regularized relaxed transport, we show that the relaxation is tight and that the transport plan is an assignment. In the general case, the regularization prevents the solution from being an assignment, but we show that the corresponding map can be used to solve imaging problems. We show an illustrative application of this discrete regularized transport to color transfer between images. This imaging problem cannot be solved in a satisfying manner without relaxing the bijective assignment constraint because of mass variation across image color palettes. Furthermore, the regularization of the transport plan helps remove colorization artifacts due to noise amplification. We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cuturi (2013), we propose to smooth the Wasserstein distance used in the definition of Wasserstein barycenters with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms. We use these algorithms to visualize a large family of images and to solve a constrained clustering problem.