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：本文研究通过切片与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，这些方案得益于一维最优传输问题闭式解的存在性及其计算优势。尽管此类方案已在数据科学应用中成功使用，但其收敛性的理论结果尚不充分。本文的主要贡献是给出了随机切片匹配方案的几乎必然收敛性证明。该证明建立在将其解释为Wasserstein空间上随机梯度下降方案的基础上。文中同时展示了逐步图像形变的数值算例。