Measure-valued spline curves: an optimal transport viewpoint

TL;DR

This paper introduces a method to smoothly interpolate probability measures by extending spline curves into the space of measures using optimal transport, enabling better modeling of distributions over time.

Contribution

It extends the concept of spline curves from Euclidean points to probability measures within the optimal transport framework, providing a novel approach for measure interpolation.

Findings

Develops a new measure-valued spline interpolation method

Demonstrates the approach's effectiveness on empirical probability measures

Bridges spline theory with optimal transport for measure interpolation

Abstract

The aim of this article is to introduce and address the problem to smoothly interpolate (empirical) probability measures. To this end, we lift the concept of a spline curve from the setting of points in a Euclidean space that that of probability measures, using the framework of optimal transport.