Order Constraints in Optimal Transport

TL;DR

This paper introduces order constraints into optimal transport to enhance the interpretability of solutions, providing an efficient method with proven theoretical properties and demonstrating improved explainability on NLP and image tasks.

Contribution

It presents a novel formulation of optimal transport with order constraints and an efficient solution method that scales well and is theoretically grounded.

Findings

Order constraints improve explainability in NLP and image tasks.

The proposed method scales better than standard optimal transport approaches.

Theoretical properties of the new formulation are established.

Abstract

Optimal transport is a framework for comparing measures whereby a cost is incurred for transporting one measure to another. Recent works have aimed to improve optimal transport plans through the introduction of various forms of structure. We introduce novel order constraints into the optimal transport formulation to allow for the incorporation of structure. We define an efficient method for obtaining explainable solutions to the new formulation that scales far better than standard approaches. The theoretical properties of the method are provided. We demonstrate experimentally that order constraints improve explainability using the e-SNLI (Stanford Natural Language Inference) dataset that includes human-annotated rationales as well as on several image color transfer examples.