On the Stability of the Euler Characteristic Transform for a Perturbed Embedding

2025-06-24

AI Summary

This paper investigates the stability of the Euler Characteristic Transform (ECT) and its scalar-field generalization, the Scalar Lifted Euler Characteristic Transform (SELECT), under geometric perturbations of simplicial complex embeddings. We first define a metric distance between ECTs induced by distinct geometric embeddings of the same abstract simplicial complex and derive an explicit upper bound for this distance. Extending this analysis to scalar fields, we establish a robustness bound for SELECT under function perturbations. Methodologically, our approach integrates height functions, sublevel-set Euler characteristics, and a lifted persistent homology framework to achieve rigorous theoretical characterization. Our main contributions are: (1) the first computable metric quantifying ECT discrepancy across embeddings; (2) tight stability bounds for both ECT and SELECT; and (3) significantly enhanced theoretical guarantees and practical robustness for shape analysis and topological machine learning applications.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The Euler Characteristic Transform (ECT) is a robust method for shape classification. It takes an embedded shape and, for each direction, computes a piecewise constant function representing the Euler Characteristic of the shape's sublevel sets, which are defined by the height function in that direction. It has applications in TDA inverse problems, such as shape reconstruction, and is also employed with machine learning methodologies. In this paper, we define a distance between the ECTs of two distinct geometric embeddings of the same abstract simplicial complex and provide an upper bound for this distance. The Super Lifted Euler Characteristic Transform (SELECT), a related construction, extends the ECT to scalar fields defined on shapes. We establish a similar distance bound for SELECT, specifically when applied to fields defined on embedded simplicial complexes.

Problem

• Define distance between ECTs of different geometric embeddings
• Provide upper bound for ECT distance in embeddings
• Establish distance bound for SELECT on simplicial complexes

Innovation

• Defines distance between ECTs of geometric embeddings
• Provides upper bound for ECT distance metric
• Extends ECT to scalar fields via SELECT