Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams

TL;DR

This paper introduces the Persistence Fisher kernel, a positive definite kernel for persistence diagrams based on Fisher information geometry, enabling effective machine learning analysis of topological data without approximation.

Contribution

It proposes a novel Fisher information geometry-based kernel for persistence diagrams that is positive definite and does not require approximation, along with theoretical analysis and efficient computation methods.

Findings

The PF kernel is stable and infinitely divisible.

It achieves favorable performance on benchmark datasets.

The kernel has linear time complexity for approximation.

Abstract

Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data such as shapes, linked twist maps, and material data. Among them, \textit{persistent homology} is a well-known tool to extract robust topological features, and outputs as \textit{persistence diagrams} (PDs). However, PDs are point multi-sets which can not be used in machine learning algorithms for vector data. To deal with it, an emerged approach is to use kernel methods, and an appropriate geometry for PDs is an important factor to measure the similarity of PDs. A popular geometry for PDs is the \textit{Wasserstein metric}. However, Wasserstein distance is not \textit{negative definite}. Thus, it is limited to build positive definite kernels upon the Wasserstein distance \textit{without approximation}. In this work, we rely upon the alternative \textit{Fisher information geometry} to propose a positive definite kernel for PDs \textit{without approximation}, namely the Persistence Fisher (PF) kernel. Then, we analyze eigensystem of the integral operator induced by the proposed kernel for kernel machines. Based on that, we derive generalization error bounds via covering numbers and Rademacher averages for kernel machines with the PF kernel. Additionally, we show some nice properties such as stability and infinite divisibility for the proposed kernel. Furthermore, we also propose a linear time complexity over the number of points in PDs for an approximation of our proposed kernel with a bounded error. Throughout experiments with many different tasks on various benchmark datasets, we illustrate that the PF kernel compares favorably with other baseline kernels for PDs.