Second order models for optimal transport and cubic splines on the Wasserstein space

TL;DR

This paper extends the concept of cubic spline interpolation to the Wasserstein space of probability densities, introducing new theoretical formulations and numerical methods for higher-order interpolation in optimal transport.

Contribution

It proposes a novel extension of cubic splines to Wasserstein space and introduces simplified approaches using multi-marginal and semi-discrete optimal transport techniques.

Findings

Successful formulation of cubic splines in Wasserstein space

Development of two numerical methods for higher-order interpolation

Potential applications in probabilistic modeling and data analysis

Abstract

On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural extension of cubic splines to the Wasserstein space, we propose a simpler approach based on the relaxation of the variational problem on the path space. We explore two different numerical approaches, one based on multi-marginal optimal transport and entropic regularization and the other based on semi-discrete optimal transport.