Euler Characteristic Tools For Topological Data Analysis

TL;DR

This paper introduces Euler characteristic profiles and hybrid transforms as efficient, stable topological descriptors for data analysis, achieving high performance in supervised and unsupervised tasks with low computational cost.

Contribution

It develops novel Euler characteristic-based descriptors and transforms, demonstrating their effectiveness and stability in topological data analysis.

Findings

Achieves state-of-the-art supervised performance with low computational cost

Provides efficient topological signal compression methods

Proves stability and asymptotic guarantees for the descriptors

Abstract

In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic profile. We show that this simple descriptor achieve state-of-the-art performance in supervised tasks at a very low computational cost. Inspired by signal analysis, we compute hybrid transforms of Euler characteristic profiles. These integral transforms mix Euler characteristic techniques with Lebesgue integration to provide highly efficient compressors of topological signals. As a consequence, they show remarkable performances in unsupervised settings. On the qualitative side, we provide numerous heuristics on the topological and geometric information captured by Euler profiles and their hybrid transforms. Finally, we prove stability results for these descriptors as well as asymptotic guarantees in random settings.