This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure. Similar to the sliced Wasserstein distance, these schemes benefit from the availability of closed-form solutions for the one-dimensional optimal transport problem and the associated computational advantages. While such schemes have already been successfully utilized in data science applications, not too many results on their convergence are available. The main contribution of this paper is an almost sure convergence proof for stochastic slicing-and-matching schemes. The proof builds on an interpretation as a stochastic gradient descent scheme on the Wasserstein space. Numerical examples on step-wise image morphing are demonstrated as well.

翻译：本文研究通过切片-匹配过程定义的测度迁移与逼近问题的迭代方案。与切片Wasserstein距离类似，这些方案受益于一维最优传输问题封闭形式解的可获取性及其相关计算优势。尽管此类方案已在数据科学应用中取得成功，但其收敛性方面的研究成果尚不丰富。本文的主要贡献在于证明了随机切片-匹配方案的几乎必然收敛性。该证明基于将其解释为Wasserstein空间上的随机梯度下降方案。此外，文中还展示了逐步图像变形中的数值实例。