Revisiting the Sliced Wasserstein Kernel for persistence diagrams: a Figalli-Gigli approach

TL;DR

This paper introduces a new Sliced Figalli-Gigli Kernel (SFGK) for persistence diagrams, improving geometric fidelity and handling infinite diagrams, while maintaining computational efficiency and comparable performance to existing kernels.

Contribution

It proposes a kernel based on the Figalli-Gigli distance, offering a more natural geometric interpretation and extending applicability to infinite persistence diagrams.

Findings

SFGK shares key properties with SWK, including distortion bounds.

SFGK can handle infinite persistence diagrams and measures.

Numerical experiments show SFGK performs as well as SWK on benchmarks.

Abstract

The Sliced Wasserstein Kernel (SWK) for persistence diagrams was introduced in (Carri{\`e}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{\`e}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.