Convergence Rates for Distribution Matching with Sliced Optimal Transport

TL;DR

The paper tackles distribution matching via sliced optimal transport, focusing on the slice-matching scheme as an iterative, stochastic gradient-like method. It develops quantitative convergence rates by establishing random Polyak–Łojasiewicz inequalities for the Sliced-Wasserstein objective and shows that, in the Gaussian/elliptic setting with random orthonormal directional updates, the trajectory constants can be controlled through covariance eigenvalues, yielding explicit rates like $\mathbb{E}[\mathscr{F}(\sigma_k)] \lesssim k^{-(2\alpha-1)}$ for $\alpha\in(\tfrac{1}{2},1)$. The analysis combines static PL inequalities for bounded densities with a trajectory-wise eigenvalue control on the covariance under the Gaussian structure (Bures-Wasserstein geometry), providing both theoretical guarantees and practical insights. Numerical experiments corroborate the theory, revealing faster convergence with orthonormal-basis sampling and revealing dimension and step-size effects consistent with the derived rates, while extending experiments to non-Gaussian targets indicates broader applicability and open directions for regularization to maintain regularity along dynamics.

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.

Figures (9)

• Figure 1: Evolution of $SW_2^2(\sigma_k, \mu)$ when $\sigma =\mathcal{N}(0,\Sigma)$ and $\mu=\mathcal{N}(0,{\bf I }_d)$
• Figure 2: Minimum and maximum eigenvalues of $\Sigma_k$ when $\sigma =\mathcal{N}(0,\Sigma)$ and $\mu=\mathcal{N}(0,{\bf I}_d)$
• Figure 3: Comparison of sampling strategies: single direction $\theta_{k+1}$ or orthonormal basis $P_{k+1}$. We report $\lambda_{\min}(\Sigma_k)$ and $\lambda_{\max}(\Sigma_k)$ with $\sigma =\mathcal{N}(0,\Sigma)$, $\mu=\mathcal{N}(0,{\bf I}_d)$, $d = 5$.
• Figure 4: Evolution of $SW_2^2(\sigma_k, \mu)$ for discrete source and target distributions. The source and target samples are distributed from Gaussian mixtures.
• Figure 5: Evolution of $SW_2^2(\sigma_k, \mu)$ when $\sigma =\mathcal{N}(0,\Sigma)$ and $\mu=\mathcal{N}(0,\Lambda)$

Theorems & Definitions (37)

• Lemma 3.1: li2023measure, Lemma A.1
• Proposition 3.2
• Proposition 3.3
• Theorem 4.1
• Theorem 4.2
• Proposition 4.3
• Proposition 4.4: PL inequality on $\mathcal{G}_{m,M}$
• Remark 4.5: Centered Gaussians
• Remark 4.6: Elliptically contoured distributions
• Definition A.1