Covariance-modulated optimal transport and gradient flows

TL;DR

This paper introduces a covariance-modulated optimal transport framework that improves convergence rates in gradient flows, with applications to ensemble Kalman methods for inverse problems.

Contribution

It develops a new modulated transport metric that splits into mean-covariance and shape components, enhancing convergence analysis and dynamics understanding.

Findings

Transport metric induces favorable geometric properties.

Gradient flows exhibit exponential convergence independent of Gaussian targets.

Splitting into moments and shape evolution clarifies the dynamics.

Abstract

We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.