Fast Unbalanced Optimal Transport on a Tree

TL;DR

This paper analyzes the computational complexity of unbalanced optimal transport problems, proving certain cases are hard and providing a fast quasi-linear time algorithm for tree metrics, enabling large-scale applications.

Contribution

It establishes complexity bounds for unbalanced optimal transport and introduces a novel efficient algorithm for tree metrics, facilitating large-scale data analysis.

Findings

Kantorovich Rubinstein distance cannot be computed in strongly subquadratic time under ETH.

Proposed algorithm solves unbalanced optimal transport on trees in quasi-linear time.

Algorithm processes one million nodes in less than one second.

Abstract

This study examines the time complexities of the unbalanced optimal transport problems from an algorithmic perspective for the first time. We reveal which problems in unbalanced optimal transport can/cannot be solved efficiently. Specifically, we prove that the Kantorovich Rubinstein distance and optimal partial transport in the Euclidean metric cannot be computed in strongly subquadratic time under the strong exponential time hypothesis. Then, we propose an algorithm that solves a more general unbalanced optimal transport problem exactly in quasi-linear time on a tree metric. The proposed algorithm processes a tree with one million nodes in less than one second. Our analysis forms a foundation for the theoretical study of unbalanced optimal transport algorithms and opens the door to the applications of unbalanced optimal transport to million-scale datasets.