Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems

TL;DR

This paper introduces Euler Characteristic Curves and Profiles as stable, computationally efficient shape invariants for big data analysis, overcoming some limitations of persistent homology.

Contribution

It proposes Euler characteristic-based invariants for multi-parameter filtrations, with efficient distributed algorithms and demonstrated stability and practical utility.

Findings

Efficient algorithms for computing Euler Curves and Profiles in distributed settings.

Euler-based invariants are stable and robust for big data shape analysis.

Practical applications demonstrate their effectiveness in real-world data problems.

Abstract

Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to distribute, it is hard to generalize to multifiltrations and is computationally prohibitive for big data-sets. In this paper we study the concept of Euler Characteristics Curves, for one parameter filtrations and Euler Characteristic Profiles, for multi-parameter filtrations. While being a weaker invariant in one dimension, we show that Euler Characteristic based approaches do not possess some handicaps of persistent homology; we show efficient algorithms to compute them in a distributed way, their generalization to multifiltrations and practical applicability for big data problems. In addition we show that the Euler Curves and Profiles enjoys certain type of stability which makes them robust tool in data analysis. Lastly, to show their practical applicability, multiple use-cases are considered.