Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertions
This provides the first eigenvalue-level robustness guarantee for spectral topological data analysis, ensuring stability under local updates and enabling reliable error control in dynamic data settings, which is important for applications in biology, physics, and machine learning.
The paper tackled the problem of precisely bounding the change in a single eigenvalue of a persistent Laplacian when one simplex is added, which was previously unknown, and proved a uniform Lipschitz bound that limits the variation to at most twice the Euclidean norm of the simplex's boundary.
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.