Euler Characteristic Tools For Topological Data Analysis

TL;DR

This work studies Euler characteristic-based descriptors for topological data analysis, introducing Euler characteristic profiles $\chi_{\mathcal{F}}$ and their hybrid transforms $\psi_{\mathcal{F}}^{\kappa}$ as computationally efficient alternatives to persistence diagrams in multi-parameter settings. By exploiting one-critical filtrations and kernel-based transforms, the authors demonstrate strong predictive performance in supervised tasks, effective information compression in unsupervised tasks, and robust stability guarantees against perturbations, along with limit theorems for i.i.d. samples and multi-parameter filtrations. The methodology is validated on curvature regression, ORBIT5K, Sydney object recognition, and graph datasets, showing competitive accuracy with significant speedups and the ability to handle up to five filtration parameters. The results offer a practical, scalable framework for topological learning that preserves essential multi-parameter information while enabling rapid computation and theoretical guarantees for both statistical and stability properties.

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.

Figures (12)

• Figure 1: Balls with varying radius $t>0$ centered at each point of a finite subset $\mathbb{X}\subseteq {\mathbb R}^2$. These balls are used to define the Čech filtration $\check{\mathcal{C}}(\mathbb{X})$ and its corresponding persistence diagrams of dimension 0 (in red) and 1 (in blue).
• Figure 2: A finitely generated $2$-parameter filtration (a) and its associated Euler characteristic surface (b). All vertices have one birth time, while all other simplices have two.
• Figure 3: Hybrid transforms of $\chi_{\mathcal{F}}= \mathbf{1}_{[a,b)}$ for several choices of kernel $\kappa$
• Figure 4: Examples of alpha complexes on PPP and GPP point clouds at two scales $t_1$ (Figures (a) and (b)) and $t_2$ (Figures (c) and (d)) with $t_1 < t_2$.
• Figure 5: Euler characteristic curves and their transforms for PPP VS GPP data set

Theorems & Definitions (23)

• Example 1
• Example 2
• Example 3
• Definition 1
• Definition 2
• Lemma 3
• Definition 4
• Lemma 5
• Remark 6
• Example 4