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.

On the shape that matters – topology and geometry in data science

“…In such cases, simplex s will contribute a value of 1 at every point that is coordinate-wise greater or equal to any of f 1 (s), …, f k (s), and 0 at all other points. Through this method, we obtain a stable invariant of an n-dimensional filtration, referred to as the Euler characteristic profile, see [6] for further discussion and properties.…”

“…Let us consider a simple example of such a scenario: a problem of analysing prostate cancer features on hematoxylin and eosin (H&E) stained slide images. Our results are based on publicly available 5182 images, each of 512 × 512 resolution, obtained from EMS MAGAZINE 132 (2024) the Open Science Framework [11], as analysed in [6]. These images represent various regions of interest (ROIs) from prostate cancer H&E slices, collected from 39 patients.…”