Stability and Inference of the Euler Characteristic Transform

Area of Science:

• Topological Data Analysis (TDA)
• Differential Geometry
• Computational Geometry

Background:

• The Euler characteristic transform (ECT) is a signature in topological data analysis for shape summarization.
• While fast and injective for many shapes, the ECT is sensitive to small perturbations.
• Existing methods often rely on triangulation size, which can be unstable.

Purpose of the Study:

• To develop a new metric for assessing the stability of the ECT on one-dimensional shapes.
• To demonstrate that the ECT is stable with respect to curvature, not just triangulation size.
• To create a statistically sound and computationally efficient estimator for the ECT.

Main Methods:

• Introduction of a novel metric for compact one-dimensional shapes.
• Proof of ECT stability using this new metric, focusing on intrinsic shape properties (curvature).
• Development of a Gaussian process-based statistical estimator for the ECT.

Main Results:

• The proposed metric establishes the stability of the ECT for one-dimensional shapes.
• Curvature is identified as a key factor in controlling ECT stability, offering an advantage over mesh-dependent measures.
• A consistent statistical estimator for the ECT is developed, converging to the true ECT with increasing sample size.

Conclusions:

• The ECT can be made robust to perturbations through a curvature-based stability analysis.
• The developed estimator provides a reliable tool for analyzing noisy shape data in TDA.
• This work advances the practical application of TDA by improving the reliability of shape signatures.