Convergence Rates for Distribution Matching with Sliced Optimal Transport

2026-02-11

AI Summary

This work investigates the non-asymptotic convergence rates of distribution matching algorithms based on Sliced Optimal Transport. By establishing a Łojasiewicz-type inequality for the Sliced-Wasserstein objective function, the study provides the first rigorous theoretical guarantee of non-asymptotic convergence for such algorithms. In the Gaussian setting, the constants in the inequality are explicitly characterized in terms of the optimization trajectory. The analysis further reveals the critical role of random orthogonal basis sampling in ensuring algorithmic stability and quantifies the dependence of the convergence rate on both dimensionality and step size. Numerical experiments corroborate the theoretical predictions, particularly demonstrating the effectiveness of orthogonal basis sampling in enhancing stability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We study the slice-matching scheme, an efficient iterative method for distribution matching based on sliced optimal transport. We investigate convergence to the target distribution and derive quantitative non-asymptotic rates. To this end, we establish __ojasiewicz-type inequalities for the Sliced-Wasserstein objective. A key challenge is to control along the trajectory the constants in these inequalities. We show that this becomes tractable for Gaussian distributions. Specifically, eigenvalues are controlled when matching along random orthonormal bases at each iteration. We complement our theory with numerical experiments and illustrate the predicted dependence on dimension and step-size, as well as the stabilizing effect of orthonormal-basis sampling.

Problem

• distribution matching
• sliced optimal transport
• convergence rates
• Sliced-Wasserstein
• non-asymptotic analysis

Innovation

• Sliced Optimal Transport
• Convergence Rates
• Łojasiewicz Inequality
• Distribution Matching
• Orthonormal Basis Sampling