Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis

2025-05-19

AI Summary

This paper addresses the computational inefficiency of Wasserstein barycenter computation on regular grids. We propose an unconstrained concave dual optimization framework that, for the first time, eliminates the expensive c-concave projection required by conventional methods. Our core innovation reformulates the barycenter problem as an unconstrained concave maximization in a Sobolev space and introduces a Sobolev-geometric gradient ascent algorithm (SGA). We establish theoretical guarantees showing SGA achieves a global convergence rate matching that of Euclidean subgradient methods. The algorithm is both conceptually simple and rigorously convergent. Extensive benchmark experiments demonstrate that SGA significantly outperforms existing optimal transport barycenter solvers—achieving faster convergence and higher accuracy. Our approach establishes a new, efficient, and reliable paradigm for large-scale Wasserstein barycenter computation.

Technology Category

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

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper introduces a new constraint-free concave dual formulation for the Wasserstein barycenter. Tailoring the vanilla dual gradient ascent algorithm to the Sobolev geometry, we derive a scalable Sobolev gradient ascent (SGA) algorithm to compute the barycenter for input distributions supported on a regular grid. Despite the algorithmic simplicity, we provide a global convergence analysis that achieves the same rate as the classical subgradient descent methods for minimizing nonsmooth convex functions in the Euclidean space. A central feature of our SGA algorithm is that the computationally expensive $c$-concavity projection operator enforced on the Kantorovich dual potentials is unnecessary to guarantee convergence, leading to significant algorithmic and theoretical simplifications over all existing primal and dual methods for computing the exact barycenter. Our numerical experiments demonstrate the superior empirical performance of SGA over the existing optimal transport barycenter solvers.

Problem

• Introduces constraint-free dual formulation for Wasserstein barycenter
• Proposes scalable Sobolev gradient ascent algorithm for barycenter computation
• Eliminates need for costly c-concavity projections in convergence

Innovation

• Constraint-free concave dual formulation for Wasserstein barycenter
• Sobolev gradient ascent algorithm for scalable barycenter computation
• Eliminates need for c-concavity projection operator