Learning Cost Functions for Optimal Transport
This work addresses a computational bottleneck in inverse optimal transport, which is important for applications in machine learning and data science where cost functions need to be inferred from data.
The paper tackles the inverse optimal transport problem of learning cost functions from observed transport plans, proposing an unconstrained convex optimization formulation with customizable regularization and developing two efficient numerical algorithms. Numerical results show promising efficiency and accuracy advantages over state-of-the-art methods.
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.