# Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertions

arXiv: https://arxiv.org/abs/2506.21352 · HTML: https://mlsanity.com/paper/2506.21352/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 20 references that are in the map.
- First posted on arXiv 26 Jun 2025.

## Builds on (most cited first, 10 of 10)

- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Persistent Laplacians: properties, algorithms and implications](https://mlsanity.com/paper/2012.02808/) (2020)
- [Recovering Manifold Structure Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2410.01149/) (2024)
- [PLPCA: Persistent Laplacian Enhanced-PCA for Microarray Data Analysis](https://mlsanity.com/paper/2306.06292/) (2023)
- [Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning](https://mlsanity.com/paper/2408.00220/) (2024)
- [A generalization of the persistent Laplacian to simplicial maps](https://mlsanity.com/paper/2302.03771/) (2023)
- [Persistent Laplacian-enhanced Algorithm for Scarcely Labeled Data Classification](https://mlsanity.com/paper/2305.16239/) (2023)
- [SE(3)-Invariant Multiparameter Persistent Homology for Chiral-Sensitive Molecular Property Prediction](https://mlsanity.com/paper/2312.07633/) (2023)
- [Quantum Distance Approximation for Persistence Diagrams](https://mlsanity.com/paper/2402.17295/) (2024)

## Abstract (arXiv)

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.