Embedding optimal transports in statistical manifolds

TL;DR

This paper introduces a universal geometric framework for optimal transport problems with convex cost functions derived from cumulant generating functions, linking them to information geometry of exponential families.

Contribution

It shows that extended gradient maps in optimal transport can be represented as supergradients of a concave function on probability distributions, unifying the geometry of these problems.

Findings

Extended gradient maps are supergradients of a concave function.

Provides a universal geometric embedding for optimal transports.

Connects optimal transport with information geometry of exponential families.

Abstract

We consider Monge-Kantorovich optimal transport problems on $\mathbb{R}^d$, $d\ge 1$, with a convex cost function given by the cumulant generating function of a probability measure. Examples include the Wasserstein-2 transport whose cost function is the square of the Euclidean distance and corresponds to the cumulant generating function of the multivariate standard normal distribution. The optimal coupling is usually described via an extended notion of convex/concave functions and their gradient maps. These extended notions are nonintuitive and do not satisfy useful inequalities such as Jensen's inequality. Under mild regularity conditions, we show that all such extended gradient maps can be recovered as the usual supergradients of a nonnegative concave function on the space of probability distributions. This embedding provides a universal geometry for all such optimal transports and an unexpected connection with information geometry of exponential families of distributions.