Convergence Rates for Distribution Matching with Sliced Optimal Transport

TL;DR

This paper analyzes the convergence behavior of a sliced optimal transport-based distribution matching method, deriving quantitative rates and demonstrating their dependence on distribution properties and algorithm parameters.

Contribution

It establishes non-asymptotic convergence rates for the slice-matching scheme using new inequalities and eigenvalue controls, especially for Gaussian distributions.

Findings

Convergence rates depend on distribution properties and step-size.

Orthonormal-basis sampling stabilizes the matching process.

Numerical experiments confirm theoretical predictions.

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.