A Kernel for Multi-Parameter Persistent Homology

TL;DR

This paper introduces a new kernel for multi-parameter persistent homology that is stable, efficiently computable, and bridges topological data analysis with machine learning for complex multivariate data.

Contribution

It proposes a novel kernel construction for multi-parameter persistence by integrating a one-parameter kernel along lines, establishing a theoretical connection with machine learning.

Findings

Kernel is stable under data perturbations

Kernel can be computed efficiently

Connects topological data analysis with machine learning for multivariate data

Abstract

Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis.