# Unnormalized Optimal Transport

## TL;DR

This paper extends optimal transport theory to handle unnormalized and unequal masses, introducing new equations and duality formulas that enable efficient computation of distances between unnormalized densities.

## Contribution

It develops a novel extension of the Monge-Kantorovich problem for unnormalized masses, including new equations and duality formulas, with efficient solution methods.

## Key findings

- Introduces a new Monge-Ampere type equation.
- Develops a Kantorovich duality formula for unnormalized masses.
- Provides an efficient computational approach using primal-dual algorithms.

## Abstract

We propose an extension of the computational fluid mechanics approach to the Monge-Kantorovich mass transfer problem, which was developed by Benamou-Brenier. Our extension allows optimal transfer of unnormalized and unequal masses. We obtain a one-parameter family of simple modifications of the formulation in [4]. This leads us to a new Monge-Ampere type equation and a new Kantorovich duality formula. These can be solved efficiently by, for example, the Chambolle-Pock primal-dual algorithm. This solution to the extended mass transfer problem gives us a simple metric for computing the distance between two unnormalized densities. The L1 version of this metric was shown in [23] (which is a precursor of our work here) to have desirable properties.