Fast Computation of Free-Support Wasserstein Medians

2026-06-17

AI Summary

This work addresses the computational inefficiency of exact empirical Wasserstein median computation, which traditionally relies on nested optimization. The authors propose a fixed-weight, free-support direct solver that eliminates inner loops by solving an exact optimal transport subproblem, performing barycentric projection, and updating support points at each iteration. A key innovation is the use of an inverse-distance weighted averaging strategy to update support points, which yields a tight majorization-minimization (MM) surrogate function. This construction guarantees monotonic descent of the objective, preserves the convex hull of the support set, and ensures finite-time convergence. Experiments demonstrate that the method achieves objective values comparable to the nested Weiszfeld algorithm with significantly fewer optimal transport calls and exhibits superior robustness over Wasserstein barycenters in contaminated data, posterior aggregation, and image prototyping tasks.

Abstract

The Wasserstein median is a robust alternative to the Wasserstein barycenter for averaging probability measures, but exact empirical computation can be expensive. A natural metric-space Weiszfeld scheme updates the current candidate by solving a weighted Wasserstein barycenter problem at each outer iteration, producing a nested optimization problem. We propose a direct fixed-weight free-support solver that avoids this inner barycenter loop. At each iteration, the method solves exact optimal transport (OT) subproblems from the current candidate to the input measures, computes barycentric projections of the selected plans, and relocates each support atom to an inverse-distance-weighted average of its projected destinations. For a smoothed median objective, we show that this relocation is the exact minimizer of a tight majorization--minimization surrogate. This yields monotone descent for exact transport subproblems, convex-hull invariance, a finite-time best-residual rate, residual-to-gradient control under differentiability, and fixed-point and stationarity characterizations. We also give smoothing, stability, and resolution-consistency results clarifying the fixed-weight approximation. In exact-OT benchmarks, the direct solver attains median objectives close to tightly solved nested Weiszfeld baselines while using substantially fewer exact transport subproblems. Additional contamination, posterior aggregation, and image-prototype experiments show that the direct solver produces median summaries comparable to nested computation and less sensitive to outlying distributions than Wasserstein barycenters.

Problem

• Wasserstein median
• free-support
• computational efficiency
• optimal transport
• nested optimization

Innovation

• Wasserstein median
• free-support
• optimal transport
• majorization-minimization
• robust averaging