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.

翻译：本文研究切片匹配方案——一种基于切片最优传输的高效迭代式分布匹配方法。我们探讨了该方法向目标分布的收敛特性，并推导了定量的非渐近收敛速率。为此，我们为切片Wasserstein目标函数建立了Łojasiewicz型不等式。一个关键挑战在于控制这些不等式沿优化轨迹的常数项。我们证明该问题对于高斯分布具有可处理性：具体而言，当在每次迭代中沿随机正交基进行匹配时，特征值能够得到有效控制。我们通过数值实验补充理论分析，展示了维度与步长对收敛速率的预测性依赖关系，以及正交基采样的稳定化效应。