# Persistent Laplacians: properties, algorithms and implications

arXiv: https://arxiv.org/abs/2012.02808 · HTML: https://mlsanity.com/paper/2012.02808/

## Key facts

- Cited by 28 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 4 Dec 2020.

## Built on by (most cited first, 28 of 28)

- [Understanding Oversquashing in GNNs through the Lens of Effective Resistance](https://mlsanity.com/paper/2302.06835/) (2023)
- [Cayley Graph Propagation](https://mlsanity.com/paper/2410.03424/) (2024)
- [CCP: Correlated Clustering and Projection for Dimensionality Reduction](https://mlsanity.com/paper/2206.04189/) (2022)
- [Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal](https://mlsanity.com/paper/2311.14427/) (2023)
- [PLPCA: Persistent Laplacian Enhanced-PCA for Microarray Data Analysis](https://mlsanity.com/paper/2306.06292/) (2023)
- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [ChatGPT for Computational Topology](https://mlsanity.com/paper/2310.07570/) (2023)
- [Analyzing Single Cell RNA Sequencing with Topological Nonnegative Matrix Factorization](https://mlsanity.com/paper/2310.15744/) (2023)
- [Persistent Homology for High-dimensional Data Based on Spectral Methods](https://mlsanity.com/paper/2311.03087/) (2023)
- [Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning](https://mlsanity.com/paper/2408.00220/) (2024)
- [Multiscale methods for signal selection in single-cell data](https://mlsanity.com/paper/2206.07760/) (2022)
- [Persistent Laplacian-enhanced Algorithm for Scarcely Labeled Data Classification](https://mlsanity.com/paper/2305.16239/) (2023)
- [K-Nearest-Neighbors Induced Topological PCA for scRNA Sequence Data Analysis](https://mlsanity.com/paper/2310.14521/) (2023)
- [Provable quantum speedups for computing persistence in topological data analysis](https://mlsanity.com/paper/2410.21258/) (2024)
- [Quantum Distance Approximation for Persistence Diagrams](https://mlsanity.com/paper/2402.17295/) (2024)
- [Linguistics from a topological viewpoint](https://mlsanity.com/paper/2403.15440/) (2024)
- [Graph Persistence goes Spectral](https://mlsanity.com/paper/2506.06571/) (2025)
- [Multi-dimensional Persistent Sheaf Laplacians for Image Analysis](https://mlsanity.com/paper/2602.14846/) (2026)
- [A Persistent Homology Design Space for 3D Point Cloud Deep Learning](https://mlsanity.com/paper/2604.04299/) (2026)
- [Mathematics-assisted directed evolution and protein engineering](https://mlsanity.com/paper/2306.04658/) (2023)
- [Higher-order Cross-structural Embedding Model for Time Series Analysis](https://mlsanity.com/paper/2410.22984/) (2024)
- [A Quotient Homology Theory of Representation in Neural Networks](https://mlsanity.com/paper/2502.01360/) (2025)
- [Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertions](https://mlsanity.com/paper/2506.21352/) (2025)
- [Persistent Sheaf Laplacian Analysis of Protein Stability and Solubility Changes upon Mutation](https://mlsanity.com/paper/2601.12219/) (2026)
- [Persistent Nonnegative Matrix Factorization via Multi-Scale Graph Regularization](https://mlsanity.com/paper/2602.22536/) (2026)
- [A Hierarchical Sheaf Spectral Embedding Framework for Single-Cell RNA-seq Analysis](https://mlsanity.com/paper/2603.26858/) (2026)
- [Analytic Torsion and Spectral Gap Capture Persistent-Laplacian Performance](https://mlsanity.com/paper/2606.16990/) (2026)
- [Collapsed Effective Operators for Higher-order Structures](https://mlsanity.com/paper/2606.23517/) (2026)

## Abstract (arXiv)

We present a thorough study of the theoretical properties and devise efficient algorithms for the persistent Laplacian, an extension of the standard combinatorial Laplacian to the setting of pairs (or, in more generality, sequences) of simplicial complexes K hookrightarrow L, which was independently introduced by Lieutier et al. and by Wang et al. In particular, in analogy with the non-persistent case, we first prove that the nullity of the q-th persistent Laplacian DeltaqK,L equals the q-th persistent Betti number of the inclusion (K hookrightarrow L). We then present an initial algorithm for finding a matrix representation of DeltaqK,L, which itself helps interpret the persistent Laplacian. We exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix which has several important implications. In the graph case, it both uncovers a link with the notion of effective resistance and leads to a persistent version of the Cheeger inequality. This relationship also yields an additional, very simple algorithm for finding (a matrix representation of) the q-th persistent Laplacian which in turn leads to a novel and fundamentally different algorithm for computing the q-th persistent Betti number for a pair (K,L) which can be significantly more efficient than standard algorithms. Finally, we study persistent Laplacians for simplicial filtrations and present novel stability results for their eigenvalues. Our work brings methods from spectral graph theory, circuit theory, and persistent homology together with a topological view of the combinatorial Laplacian on simplicial complexes.