# Convergence Rates for Distribution Matching with Sliced Optimal Transport

arXiv: https://arxiv.org/abs/2602.10691 · HTML: https://mlsanity.com/paper/2602.10691/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 73 references that are in the map.
- First posted on arXiv 11 Feb 2026.

## Builds on (most cited first, 30 of 39)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [Sliced Wasserstein Generative Models](https://mlsanity.com/paper/1904.05408/) (2019)
- [Statistical optimal transport](https://mlsanity.com/paper/2407.18163/) (2024)
- [On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient](https://mlsanity.com/paper/1803.06573/) (2018)
- [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019)
- [Sliced Iterative Normalizing Flows](https://mlsanity.com/paper/2007.00674/) (2020)
- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Minimax Confidence Intervals for the Sliced Wasserstein Distance](https://mlsanity.com/paper/1909.07862/) (2019)
- [On Sampling with Approximate Transport Maps](https://mlsanity.com/paper/2302.04763/) (2023)
- [First-order Conditions for Optimization in the Wasserstein Space](https://mlsanity.com/paper/2209.12197/) (2022)
- [Expected Sliced Transport Plans](https://mlsanity.com/paper/2410.12176/) (2024)
- [On minimax density estimation via measure transport](https://mlsanity.com/paper/2207.10231/) (2022)
- [Triangular Flows for Generative Modeling: Statistical Consistency, Smoothness Classes, and Fast Rates](https://mlsanity.com/paper/2112.15595/) (2021)
- [Nonparametric Generative Modeling with Conditional Sliced-Wasserstein Flows](https://mlsanity.com/paper/2305.02164/) (2023)
- [Properties of Discrete Sliced Wasserstein Losses](https://mlsanity.com/paper/2307.10352/) (2023)
- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)
- [Progressive Entropic Optimal Transport Solvers](https://mlsanity.com/paper/2406.05061/) (2024)
- [A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems](https://mlsanity.com/paper/1810.13117/) (2018)
- [Long-Time Asymptotics of the Sliced-Wasserstein Flow](https://mlsanity.com/paper/2405.06313/) (2024)
- [A User's Guide to Sampling Strategies for Sliced Optimal Transport](https://mlsanity.com/paper/2502.02275/) (2025)

## Built on by (most cited first, 2 of 2)

- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)
- [Distribution Steering via Sliced Optimal Transport Control](https://mlsanity.com/paper/2608.12828/) (2026)

## Abstract (arXiv)

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 Lojasiewicz-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.