Multiplicative Schr\"odinger problem and the Dirichlet transport

TL;DR

This paper introduces a novel optimal transport framework on the simplex using Dirichlet processes, linking large deviation limits of particle systems to entropy and cost convexity, with broad applicability.

Contribution

It establishes a new connection between Dirichlet processes, Schrödinger problems, and optimal transport on the simplex, extending existing methods without gamma convergence.

Findings

Optimal transport characterized by gradients of exponentially concave functions.

Entropy plus transport cost is convex along the transport paths.

Dimension-free bounds of transport cost in terms of entropy obtained.

Abstract

We consider an optimal transport problem on the unit simplex whose solutions are given by gradients of exponentially concave functions and prove two main results. First, we show that the optimal transport is the large deviation limit of a particle system of Dirichlet processes transporting one probability measure on the unit simplex to another by coordinatewise multiplication and normalizing. The structure of our Lagrangian and the appearance of the Dirichlet process relate our problem closely to the entropic measure on the Wasserstein space as defined by von-Renesse and Sturm in the context of Wasserstein diffusion. The limiting procedure is a triangular limit where we allow simultaneously the number of particles to grow to infinity while the `noise' tends to zero. The method, which generalizes easily to many other cost functions, including the squared Euclidean distance, provides a novel combination of the Schr\"odinger problem approach due to C. L\'eonard and the related Brownian particle systems by Adams et al.which does not require gamma convergence. Second, we analyze the behavior of entropy along the paths of transport. The reference measure on the simplex is taken to be the Dirichlet measure with all zero parameters which relates to the finite-dimensional distributions of the entropic measure. The interpolating curves are not the usual McCann lines. Nevertheless we show that entropy plus a multiple of the transport cost remains convex, which is reminiscent of the semiconvexity of entropy along lines of McCann interpolations in negative curvature spaces. We also obtain, under suitable conditions, dimension-free bounds of the optimal transport cost in terms of entropy.