提出切片最优传输的收敛速率分析，揭示其在高维下的稳定机制。
Convergence Rates for Distribution Matching with Sliced Optimal Transport
- 基于切片最优传输设计迭代匹配方法，利用正交基采样控制收敛过程
- 证明了高斯分布下收敛速率与维度、步长的定量关系，具非渐近性
- 适用于高维概率分布匹配任务，尤其适合需稳定收敛的生成建模场景
我们研究了一种基于切片最优传输的高效迭代分布匹配方法——切片匹配。通过建立切片-Wasserstein 目标函数的 Lojasiewicz 型不等式，我们推导出其收敛到目标分布的定量非渐近速率。关键挑战在于沿轨迹控制这些不等式中的常数。我们证明：当每轮迭代中沿随机正交基进行匹配时，该问题可解，且特征值可被有效控制。数值实验验证了理论预测的维度与步长依赖关系，并展示了正交基采样带来的稳定性提升。
原文摘要 · Abstract (English)
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.
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