# Scalable Unbalanced Sobolev Transport for Measures on a Graph

## TL;DR

This paper introduces a scalable unbalanced Sobolev transport method for measures on a graph, addressing computational complexity and mass imbalance issues in optimal transport, with theoretical and empirical validation.

## Contribution

It extends Sobolev transport to unbalanced measures on graphs, providing a closed-form formula, geometric insights, and kernel design for efficient comparison of measures.

## Key findings

- Fast computation of the proposed UST method
- Comparable performance to existing transport baselines
- Effective kernel design for unbalanced measures

## Abstract

Optimal transport (OT) is a popular and powerful tool for comparing probability measures. However, OT suffers a few drawbacks: (i) input measures required to have the same mass, (ii) a high computational complexity, and (iii) indefiniteness which limits its applications on kernel-dependent algorithmic approaches. To tackle issues (ii)--(iii), Le et al. (2022) recently proposed Sobolev transport for measures on a graph having the same total mass by leveraging the graph structure over supports. In this work, we consider measures that may have different total mass and are supported on a graph metric space. To alleviate the disadvantages (i)--(iii) of OT, we propose a novel and scalable approach to extend Sobolev transport for this unbalanced setting where measures may have different total mass. We show that the proposed unbalanced Sobolev transport (UST) admits a closed-form formula for fast computation, and it is also negative definite. Additionally, we derive geometric structures for the UST and establish relations between our UST and other transport distances. We further exploit the negative definiteness to design positive definite kernels and evaluate them on various simulations to illustrate their fast computation and comparable performances against other transport baselines for unbalanced measures on a graph.