Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis
This work addresses the computational bottleneck in optimal transport for applications like machine learning and data analysis, offering a more efficient and simplified method for barycenter optimization.
The paper tackles the problem of computing Wasserstein barycenters by introducing a constraint-free concave dual formulation and a scalable Sobolev gradient ascent (SGA) algorithm, achieving the same convergence rate as classical methods while eliminating the need for computationally expensive projections, with numerical experiments showing superior performance over existing solvers.
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.