Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems
This provides a more scalable and generalizable shape invariant for researchers and practitioners dealing with big data in topological data analysis, though it is incremental as it builds on existing Euler characteristic concepts.
The paper tackles the computational limitations of persistent homology in topological data analysis by introducing Euler Characteristic Curves and Profiles, showing efficient distributed algorithms and practical applicability for big data problems.
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.