Euler characteristic curves and profiles: a stable shape invariant for big 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, and it is hard to generalize to multifiltrations and is computationally prohibitive for big datasets. In this article, we study the concept of Euler characteristics curves for 1-parameter filtrations and Euler characteristic profiles for multiparameter 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 enjoy a certain type of stability, which makes them robust tools for data analysis. Lastly, to show their practical applicability, multiple use cases are considered.