The Sliced Wasserstein Kernel (SWK) for persistence diagrams was introduced in (Carri{è}re et al. 2017) as a powerful tool to implicitly embed persistence diagrams in a Hilbert space with reasonable distortion. This kernel is built on the intuition that the Figalli-Gigli distance-that is the partial matching distance routinely used to compare persistence diagrams-resembles the Wasserstein distance used in the optimal transport literature, and that the later could be sliced to define a positive definite kernel on the space of persistence diagrams. This efficient construction nonetheless relies on ad-hoc tweaks on the Wasserstein distance to account for the peculiar geometry of the space of persistence diagrams. In this work, we propose to revisit this idea by directly using the Figalli-Gigli distance instead of the Wasserstein one as the building block of our kernel. On the theoretical side, our sliced Figalli-Gigli kernel (SFGK) shares most of the important properties of the SWK of Carri{è}re et al., including distortion results on the induced embedding and its ease of computation, while being more faithful to the natural geometry of persistence diagrams. In particular, it can be directly used to handle infinite persistence diagrams and persistence measures. On the numerical side, we show that the SFGK performs as well as the SWK on benchmark applications.

翻译：持久图的切片Wasserstein核（SWK）由Carrière等人于2017年提出，作为一种将持久图隐式嵌入希尔伯特空间的有效工具，并具有合理的失真度。该核函数的构建基于以下洞见：Figalli-Gigli距离（即常规用于比较持久图的部分匹配距离）类似于最优传输文献中使用的Wasserstein距离，而后者可通过切片操作在持久图空间上定义正定核函数。然而，这一高效构建依赖于对Wasserstein距离的特设调整，以适应持久图空间的特殊几何结构。本文重新审视该思路，直接采用Figalli-Gigli距离而非Wasserstein距离作为核函数的构建基础。在理论层面，我们提出的切片Figalli-Gigli核（SFGK）继承了Carrière等人SWK的大部分重要性质，包括诱导嵌入的失真度结果与计算便捷性，同时更贴合持久图的自然几何特性。特别地，该核函数可直接用于处理无限持久图与持久测度。在数值实验方面，我们证明SFGK在基准应用中的表现与SWK相当。