Wasserstein-Fisher-Rao Splines

TL;DR

This paper introduces a geometric framework for interpolating splines in the Wasserstein-Fisher-Rao space, deriving curvature and proposing an algorithm for practical computation, extending previous work on measure interpolation.

Contribution

It derives the covariant derivative and curvature in WFR space and develops a practical algorithm for spline interpolation on measures with different masses.

Findings

Derived the covariant derivative and curvature in WFR space

Proposed a practical algorithm for spline computation in WFR space

Established equivalence between geometric and Lagrangian curvature notions

Abstract

We study interpolating splines on the Wasserstein-Fisher-Rao (WFR) space of measures with differing total masses. To achieve this, we derive the covariant derivative and the curvature of an absolutely continuous curve in the WFR space. We prove that this geometric notion of curvature is equivalent to a Lagrangian notion of curvature in terms of particles on the cone. Finally, we propose a practical algorithm for computing splines extending the work of arXiv:2010.12101.