Simple unbalanced optimal transport

TL;DR

This paper presents a simple geometric model for unbalanced optimal transport using a conical extension of diffeomorphisms, revealing that mass evolves with constant acceleration along geodesics.

Contribution

It introduces a novel Riemannian framework for unbalanced optimal transport based on a conical extension, with finite-dimensional analysis and geometric comparisons.

Findings

Mass evolves with constant acceleration along geodesics.

The model provides a unified geometric perspective on unbalanced optimal transport.

Comparison with other extensions highlights unique geometric features.

Abstract

We introduce and study a simple model capturing the main features of unbalanced optimal transport. It is based on equipping the conical extension of the group of all diffeomorphisms with a natural metric, which allows a Riemannian submersion to the space of volume forms of arbitrary total mass. We describe its finite-dimensional version and present a concise comparison study of the geometry, Hamiltonian features, and geodesics for this and other extensions. One of the corollaries of this approach is that along any geodesic the total mass evolves with constant acceleration, as an object's height in a constant buoyancy field.