2023/03/23 by Lewis Marsh, Marsh, Lewis, David Beers +1 · 1 citation
Computer Science · Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Geochemistry and Geologic Mapping #Morphological variations and asymmetry #Statistics Theory (math.ST) #Topological and Geometric Data Analysis
paper · doi:10.48550/arxiv.2303.13200
openalex publication_date 2023/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Euler characteristic transform (ECT) is a signature from topological data analysis (TDA) which summarises shapes embedded in Euclidean space. Compared with other TDA methods, the ECT is fast to compute and it is injective on a broad class of shapes. However, small perturbations of a shape can lead to large distortions in its ECT. In this paper, we propose a new metric on compact one-dimensional shapes and prove that the ECT is stable with respect to this metric. Crucially, our result uses curvature, rather than the size of a triangulation of an underlying shape, to control stability. We further construct a computationally tractable statistical estimator of the ECT based on the theory of Gaussian processes. We use our stability result to prove that our estimator is consistent on shapes perturbed by independent ambient noise; i.e., the estimator converges to the true ECT as the sample size increases.