Spherical Hellinger--Kantorovich Gradient Flows

A JKO Splitting Scheme for Kantorovich-Fisher-Rao Gradient Flows

In this article we set up a splitting variant of the JKO scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge-Kantorovich distance, and then for the Hellinger/Fisher-Rao distance. Exploiting som...

Minimizing Flows for the Monge-Kantorovich Problem

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

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functi...

Optimal Transport in Competition with Reaction: The Hellinger-Kantorovich Distance and Geodesic Curves

We discuss a new notion of distance on the space of finite and nonnegative measures on Ω ⊂ Rd, which we call Hellinger–Kantorovich distance. It can be seen as an infconvolution of the well-known Kantorovich–Wasserstein distance and the HellingerKakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and...

A Gradient Descent Solution to the Monge-Kantorovich Problem

We present a new, simple, and elegant algorithm for computing the optimal mapping for the Monge-Kantorovich problem with quadratic cost. The method arises from a reformulation of the dual problem into an unconstrained minimization of a convex, continuous functional, for which the derivative can be explicitly found. The Monge-Kantorovich problem has applications in many fields; examples from ima...