Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures

Abstract

We develop a full theory for the new class of Optimal EntropyTransport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.

Contraction and regularizing properties of heat flows in metric measure spaces

“…Here we focus on the family of distances HK α depending on a tuning parameter α > 0; they correspond to the case HK α,β of [31] with the choice β := 4. In the even more specific case α = 1, HK 1 coincides with the distance HK which has been extensively studied in [32]. The general case α = 1 can be reduced to the case α = 1 by rescaling the distance d by a factor α −1/2 .…”

“…proving in particular Hellinger convergence of P t µ 0 to m as t → ∞, with exponential rate if K > 0. A second and more refined estimate involves the recently introduced family of Hellinger-Kantorovich distances HK α , α > 0, [15,14,27,31,32], which can be defined in terms of an Optimal Entropy-Transport problem [31,32]…”

{Euclidean, metric, and Wasserstein} gradient flows: an overview

“…This very interesting distance has been independently introduced by three different groups around the year 2015, and we refer to [31,32,60,63,64] for the different presentations and the different considerations which have been investigated so far. We give here only a very short description of this distance; the names chosen for it were of course different according to the different groups, and we decided to stick to the terminology chosen in [44] where it is called Kantorovich-Fisher-Rao distance to take into account all the contributions.…”

Hessian transport gradient flows

“…Currently, there are several efforts in combining both Wasserstein metric and information/Hessian metric [3,6,7,8,9,19,24,33] from various perspectives. Within the Gaussian families, several extensions are studied in [25].…”