A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

Abstract

In this paper we study the penalization of barycenters in the Wasserstein space for ϕ-exponential distributions. We obtain an explicit characterization of the barycenter in terms of the variances of the measures generalizing existing results for Gaussian measures. We then develop a gradient projection method for the computation of the barycenter establishing a Lipstchitz continuity for the gradient function. We also numerically show the influence of parameters and stability of the algorithm under small perturbation of data.PENALIZATION OF BARYCENTERS FOR ϕ-EXPONENTIAL DISTRIBUTIONS Wasserstein barycenters and optimal transports have been explored [27,18]. Several computational methods for the computation of the barycenter have been developed [12,2,21,28].Recently Wasserstein barycenters has found many applications in statistics, image processing and machine learning [29,23,31]. We refer the reader to the mentioned papers and references therein for a more detailed account of the topic.The case γ > 0 has been studied in the recent paper [8] where the existence, uniqueness and stability of a minimizer, which is called the penalized barycenter, has been established. The regularization parameter γ was proved to provide smooth barycenters especially when the input probability measures are irregular which is useful for data analysis [7,30]. In addition, the penalized barycenter problem also resembles the discretization formulation of Wasserstein gradient flows for dissipative evolution equations [17,3,10] and the fractional heat equation [14] at a given time step where {µ i } represent discretized solutions at the previous steps and γ is proportional to the time-step parameter.Gaussian measures play an important role in the study of Wasserstein barycenter problem since in this case an useful characterization of the barycenter exists [1,6] which gives rise to efficient computational algorithms such as the fixed point approach [2] and the gradient projection method [21]. Our aim in this paper is to seek for a large class of probability measures so that the penalized barycenter can be explicitly characterized and computed similarly to the case of Gaussian measures. We will study the penalization problem (1) for an important classes of probability measures, namely ϕ-exponential measures, where the entropy functional is the Tsallis entropy functional respectively. The class of ϕ-exponential measures significantly enlarges that of Gaussian measures and containing also q-Gaussian measures as special cases, cf. Section 1.3 below. To state our main results, we briefly recall the definition of ϕ-exponential measures; more detailed will be given in Section 2.1.3. ϕ-exponential distributions. Let ϕ be an increasing, positive, continuous function on (0, ∞), the ϕ-logarithmic is defined by [33] ln ϕ (t) := t 1