Probabilistic Stability Transfer for One-Dimensional Euler Characteristic Transforms

Abstract

The Euler characteristic transform (ECT) is a computable topological signature for shapes embedded in Euclidean space. Existing ECT stability results control changes in the transform through deterministic geometric perturbation bounds, but many data-generating mechanisms produce perturbations that are random and unbounded in the worst case. In such settings a uniform deterministic bound may be unavailable or too pessimistic. This paper formulates a probabilistic stability-transfer principle for one-dimensional ECT stability: a deterministic stability estimate is combined with a tail bound for the induced geometric error to obtain a high-probability ECT error bound. We instantiate this principle using the stability theorem of Marsh and Beers for embeddings of finite one-dimensional CW complexes. The resulting consequences cover ambient displacement fields conditionally on admissibility and supremum-tail estimates, and finite landmark perturbations with reconstruction. For landmark perturbations, bounded, sub-Gaussian, sub-exponential, and polynomial-tail noise assumptions give explicit high-probability ECT stability bounds. The framework isolates a geometric stochastic component that arises in shape analysis pipelines across biological morphology, medical imaging, molecular structure analysis, and remote sensing.