Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertions

TL;DR

This paper establishes a Lipschitz bound on how much persistent Laplacian eigenvalues can change when a single simplex is added, providing a stability guarantee crucial for spectral topological data analysis.

Contribution

It proves the first eigenvalue-level robustness guarantee for persistent Laplacians under local simplex insertions, independent of filtration scale and complex size.

Findings

Eigenvalues vary at most twice the boundary norm of inserted simplex.

Provides stability guarantee for spectral features in dynamic data.

Enables error control in spectral topological data analysis.

Abstract

Persistent Laplacians are matrix operators that track how the shape and structure of data transform across scales and are popularly adopted in biology, physics, and machine learning. Their eigenvalues are concise descriptors of geometric and topological features in a filtration. Although earlier work established global algebraic stability for these operators, the precise change in a single eigenvalue when one simplex, such as a vertex, edge, or triangle, is added has remained unknown. This is important because downstream tools, including heat-kernel signatures and spectral neural networks, depend directly on these eigenvalues. We close this gap by proving a uniform Lipschitz bound: after inserting one simplex, every up-persistent Laplacian eigenvalue can vary by at most twice the Euclidean norm of that simplex's boundary, independent of filtration scale and complex size. This result delivers the first eigenvalue-level robustness guarantee for spectral topological data analysis. It guarantees that spectral features remain stable under local updates and enables reliable error control in dynamic data settings.