Persistence paths and signature features in topological data analysis

TL;DR

This paper presents a novel feature map for persistent homology barcodes by converting them into paths and then computing their signatures, enabling effective statistical learning and achieving state-of-the-art classification results.

Contribution

It introduces a new method to transform barcodes into path signatures, enhancing their utility for statistical learning in topological data analysis.

Findings

Achieves state-of-the-art classification performance

Provides a universal and characteristic feature map

Enhances statistical analysis of topological features

Abstract

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations - barcode to path, path to tensor series - results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.