Kernel method for persistence diagrams via kernel embedding and weight factor

TL;DR

This paper introduces a kernel method for persistence diagrams in topological data analysis, enabling control over the influence of persistence and noise, with applications demonstrating improved performance over existing methods.

Contribution

It proposes a novel kernel for persistence diagrams that manages the effect of persistence and noise, along with a fast approximation technique.

Findings

The kernel effectively discounts noisy topological features.

The method outperforms existing kernels in practical physics data.

Provides a computationally efficient approach.

Abstract

Topological data analysis is an emerging mathematical concept for characterizing shapes in multi-scale data. In this field, persistence diagrams are widely used as a descriptor of the input data, and can distinguish robust and noisy topological properties. Nowadays, it is highly desired to develop a statistical framework on persistence diagrams to deal with practical data. This paper proposes a kernel method on persistence diagrams. A theoretical contribution of our method is that the proposed kernel allows one to control the effect of persistence, and, if necessary, noisy topological properties can be discounted in data analysis. Furthermore, the method provides a fast approximation technique. The method is applied into several problems including practical data in physics, and the results show the advantage compared to the existing kernel method on persistence diagrams.