Learning Cost Functions for Optimal Transport

TL;DR

This paper introduces a convex optimization framework for inverse optimal transport, enabling efficient learning of cost functions using novel algorithms that improve over existing methods in accuracy and computational speed.

Contribution

It presents a new convex formulation for inverse OT, along with two algorithms—one based on Sinkhorn-Knopp and another using neural networks—that avoid costly repeated forward OT computations.

Findings

Algorithms outperform state-of-the-art in efficiency

Methods achieve high accuracy in learning cost functions

Framework is flexible with customizable regularization

Abstract

Inverse optimal transport (OT) refers to the problem of learning the cost function for OT from observed transport plan or its samples. In this paper, we derive an unconstrained convex optimization formulation of the inverse OT problem, which can be further augmented by any customizable regularization. We provide a comprehensive characterization of the properties of inverse OT, including uniqueness of solutions. We also develop two numerical algorithms, one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for discrete OT, and the other one is a learning based algorithm that parameterizes the cost function as a deep neural network for continuous OT. The novel framework proposed in the work avoids repeatedly solving a forward OT in each iteration which has been a thorny computational bottleneck for the bi-level optimization in existing inverse OT approaches. Numerical results demonstrate promising efficiency and accuracy advantages of the proposed algorithms over existing state-of-the-art methods.