Stability and Inference of the Euler Characteristic Transform

TL;DR

This paper introduces a new stable metric for shapes in topological data analysis, proving the Euler characteristic transform's stability and developing a consistent statistical estimator based on Gaussian processes.

Contribution

It proposes a curvature-based metric ensuring ECT stability and constructs a Gaussian process-based estimator with proven consistency under noise.

Findings

ECT is stable under the new metric

The estimator converges to the true ECT with increasing data

Stability depends on shape curvature, not size

Abstract

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 a sufficient statistic for 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.