Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis

2023/09/06 by Mattie Ji, Ji, Mattie, Kun Meng +1
Computer Science · #46M20. Secondary: 55N31 #Algebraic Topology (math.AT) #FOS: Mathematics #Primary: 03C64 #Statistics Theory (math.ST) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2309.03142

openalex publication_date 2023/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a definable function f: S → ℝ on a definable set S, we study sublevel sets of the form S^f_t \coloneqq \x ∈ S: f(x) ≤ t\ for all t ∈ ℝ. Using o-minimal structures, we prove that the Euler characteristic of S^f_t is right-continuous with respect to t. Furthermore, when S is compact, we show that S^f_t+δ deformation retracts to S^f_t for all sufficiently small δ> 0. Applying these results, we also characterize the connections between the following concepts in topological data analysis: the Euler characteristic transform (ECT), smooth ECT, Euler-Radon transform (ERT), and smooth ERT.

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