Proximal optimal transport divergences

2025-05-17

AI Summary

Existing generative modeling frameworks struggle to unify information divergences and optimal transport (OT) distances due to their fundamentally distinct mathematical structures and computational properties. Method: We propose the Proximal OT Divergence—a novel probability divergence constructed via infimal convolution—that theoretically and computationally interpolates between information divergences and OT distances. We establish its dynamic representation, derive the associated first-order mean-field game system, and obtain a coupled nonlinear Hamilton–Jacobi–Bellman–continuity PDE system. Our analysis integrates the Benamou–Brenier framework, convex duality theory, and infimal convolution techniques. Contribution/Results: The proposed divergence is smooth, bounded, and computationally tractable, providing a rigorous foundation for OT-proximal optimization. Empirically, it significantly improves gradient stability, convergence speed, and generalization performance in distribution optimization and generative modeling tasks.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing

Abstract

We introduce proximal optimal transport divergence, a novel discrepancy measure that interpolates between information divergences and optimal transport distances via an infimal convolution formulation. This divergence provides a principled foundation for optimal transport proximals and proximal optimization methods frequently used in generative modeling. We explore its mathematical properties, including smoothness, boundedness, and computational tractability, and establish connections to primal-dual formulation and adversarial learning. Building on the Benamou-Brenier dynamic formulation of optimal transport cost, we also establish a dynamic formulation for proximal OT divergences. The resulting dynamic formulation is a first order mean-field game whose optimality conditions are governed by a pair of nonlinear partial differential equations, a backward Hamilton-Jacobi and a forward continuity partial differential equations. Our framework generalizes existing approaches while offering new insights and computational tools for generative modeling, distributional optimization, and gradient-based learning in probability spaces.

Problem

• Introduces proximal optimal transport divergence as a novel discrepancy measure
• Explores mathematical properties and computational tractability of the divergence
• Generalizes approaches for generative modeling and gradient-based learning

Innovation

• Proximal optimal transport divergence measures discrepancy
• Dynamic formulation via first order mean-field game
• Connects primal-dual formulation and adversarial learning