Quantifying the structural stability of simplicial homology

TL;DR

This paper investigates the stability of simplicial homology under perturbations by formulating a spectral matrix nearness problem and developing a bilevel optimization method, with applications to synthetic and real datasets.

Contribution

It introduces a novel approach to quantify topological stability using spectral methods and optimization, addressing a gap in understanding homology robustness.

Findings

Effective bilevel optimization procedure developed.

Method successfully applied to synthetic datasets.

Method demonstrated on transportation network data.

Abstract

The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current work, we study the stability in terms of the smallest perturbation sufficient to change the dimensionality of the corresponding homology group. Such definition requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra's homology groups. Using the resulting boundary operators, we then formulate the question of structural stability as a spectral matrix nearness problem for the corresponding higher-order graph Laplacian. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method's performance on a variety of synthetic quasi-triangulation datasets and transportation networks.