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Well-Posedness and Efficient Algorithms for Inverse Optimal Transport with Bregman Regularization

Chenglong Bao, Zanyu Li, Yunan Yang

TL;DR

This paper develops a comprehensive theory and scalable algorithm for inverse optimal transport with general Bregman regularization. It establishes existence, (equivalence-class) uniqueness, and stability of recovered costs via a forward Bregman-regularized OT map, and provides a generalized single-level convex reformulation of the inverse problem. A practical inexact block coordinate descent algorithm with linear convergence is proposed, leveraging quadratic penalties to enable diagonal-Hessian subproblems and fast Newton updates. Numerical experiments on synthetic data and a real-world marriage matching task demonstrate stable recovery and favorable computational performance, while sensitivity analyses reveal how regularization choices and parameters affect outcomes.

Abstract

This work analyzes the inverse optimal transport (IOT) problem under Bregman regularization. We establish well-posedness results, including existence, uniqueness (up to equivalence classes of solutions), and stability, under several structural assumptions on the cost matrix. On the computational side, we investigate the existence of solutions to the optimization problem with general constraints on the cost matrix and provide a sufficient condition guaranteeing existence. In addition, we propose an inexact block coordinate descent (BCD) method for the problem with a strongly convex penalty term. In particular, when the penalty is quadratic, the subproblems admit a diagonal Hessian structure, which enables highly efficient element-wise Newton updates. We establish a linear convergence rate for the algorithm and demonstrate its practical performance through numerical experiments, including the validation of stability bounds, the investigation of regularization effects, and the application to a marriage matching dataset.

Well-Posedness and Efficient Algorithms for Inverse Optimal Transport with Bregman Regularization

TL;DR

This paper develops a comprehensive theory and scalable algorithm for inverse optimal transport with general Bregman regularization. It establishes existence, (equivalence-class) uniqueness, and stability of recovered costs via a forward Bregman-regularized OT map, and provides a generalized single-level convex reformulation of the inverse problem. A practical inexact block coordinate descent algorithm with linear convergence is proposed, leveraging quadratic penalties to enable diagonal-Hessian subproblems and fast Newton updates. Numerical experiments on synthetic data and a real-world marriage matching task demonstrate stable recovery and favorable computational performance, while sensitivity analyses reveal how regularization choices and parameters affect outcomes.

Abstract

This work analyzes the inverse optimal transport (IOT) problem under Bregman regularization. We establish well-posedness results, including existence, uniqueness (up to equivalence classes of solutions), and stability, under several structural assumptions on the cost matrix. On the computational side, we investigate the existence of solutions to the optimization problem with general constraints on the cost matrix and provide a sufficient condition guaranteeing existence. In addition, we propose an inexact block coordinate descent (BCD) method for the problem with a strongly convex penalty term. In particular, when the penalty is quadratic, the subproblems admit a diagonal Hessian structure, which enables highly efficient element-wise Newton updates. We establish a linear convergence rate for the algorithm and demonstrate its practical performance through numerical experiments, including the validation of stability bounds, the investigation of regularization effects, and the application to a marriage matching dataset.

Paper Structure

This paper contains 17 sections, 15 theorems, 87 equations, 3 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Bregman Regularized Optimal Transport
  4. Well-Posedness of Inverse Bregman-Regularized OT Problems
  5. Case 1: Domain of is
  6. Case 2: Domain of is
  7. Inverse Bregman-Regularized OT as an Optimization Problem
  8. Single-Level Formulation and Existence
  9. Block Coordinate Descent for Inverse Bregman-Regularized OT
  10. Inexact BCD with Newton and Projected Gradient Steps
  11. Convergence Analysis of Inexact Block Coordinate Descent Algorithm
  12. Experiments
  13. Experiments on randomly generated marginals
  14. Verification of Stability Bound
  15. Effect of Regularization Parameter $\lambda$
  16. ...and 2 more sections

Key Result

Proposition 1

Assume the asmp:breg holds. The mapping $\mathcal{F}: \mathbb{R}_+^{n\times n} \rightarrow \mathcal{T}_\phi(\mu,\nu)$ is a surjection.

Figures (3)

  • Figure 1: Experiment results of verification of stability bounds for four type of Bregman regularizers. For the case of $\beta$-potential, we choose $\beta=0.5$.
  • Figure 2: Experiment results of effect of regularization parameter $\lambda$ for entropy and Burg entropy.
  • Figure 3: Experiment results of effect of regularization parameter $\lambda$ for Ferm-Dirac entropy and $\beta$-potential. For the case of $\beta$-potential, we choose $\beta=0.5$.

Theorems & Definitions (35)

  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Remark 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • ...and 25 more