String Diagram of Optimal Transports

TL;DR

This paper introduces a hierarchical framework for optimal transport using string diagrams, simplifying complex problems and providing efficient algorithms, including a polynomial-time solution for adversarial cases, with demonstrated experimental benefits.

Contribution

The paper presents a novel string diagram-based hierarchical framework for optimal transport, including a polynomial-time algorithm for adversarial scenarios, enhancing efficiency and modeling capabilities.

Findings

Efficient hierarchical OT synthesis via string diagrams.

Polynomial-time algorithm for adversarial cost matrix scenarios.

Experimental results show improved performance over naive methods.

Abstract

We present a novel hierarchical framework for optimal transport (OT) using string diagrams, namely string diagrams of optimal transports. This framework reduces complex hierarchical OT problems to standard OT problems, allowing efficient synthesis of optimal hierarchical transportation plans. Our approach uses algebraic compositions of cost matrices to effectively model hierarchical structures. We also study an adversarial situation with multiple choices in the cost matrices, where we present a polynomial-time algorithm for a relaxation of the problem. Experimental results confirm the efficiency and performance advantages of our proposed algorithm over the naive method.