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

2026-02-06

AI Summary

Existing sliced Wasserstein kernels for persistence diagrams rely on ad hoc modifications of the Wasserstein distance, which struggle to faithfully capture their intrinsic geometric structure. This work proposes a novel sliced kernel based on the Figalli–Gigli distance (SFGK), which for the first time constructs a positive-definite kernel directly from this metric—eliminating the need for artificial adjustments to the Wasserstein distance and naturally accommodating points with infinite persistence as well as more general persistence measures. Theoretical analysis and experiments demonstrate that SFGK achieves superior performance across multiple benchmark tasks while maintaining computational efficiency and embedding distortion comparable to existing methods. Moreover, its broader applicability significantly enhances the geometric fidelity and expressive power of kernel-based approaches for persistence diagrams.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

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.

Problem

• persistence diagrams
• Sliced Wasserstein Kernel
• Figalli-Gigli distance
• kernel methods
• topological data analysis

Innovation

• Sliced Figalli-Gigli Kernel
• Persistence Diagrams
• Figalli-Gigli Distance
• Topological Data Analysis
• Positive Definite Kernel