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.
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BibTeX
@misc{stojanovic2026probabilistic,
title = {{Probabilistic Stability Transfer for One-Dimensional Euler Characteristic Transforms}},
author = {Vanja Stojanović},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26870},
eprint = {2610.26870},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26870},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Probabilistic Stability Transfer for One-Dimensional Euler Characteristic Transforms
AU - Vanja Stojanović
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.26870
UR - https://arxiv.org/abs/2610.26870
AB - 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.
ER -