Persistent sheaf Laplacians

Abstract

The Hodge Laplacian has recently stimulated the development of graph Laplacians and cellular sheaf Laplacians. The spectral theory of these Laplacians has significantly extended the scope of algebraic topology and data analysis. Inspired by the theory of persistent Laplacians and sheaf Laplacians, this work develops the theory of persistent sheaf Laplacians for cellular sheaves. Given a persistent module of sheaf cochain complexes, one can define the notion of persistent sheaf Laplacians. Since a point cloud with a nonzero label associated with each point gives rise to a persistent module of sheaf cochain complexes, the spectra of persistent sheaf Laplacians encodes both geometrical and non-geometrical information of the given point cloud. The theory of persistent sheaf Laplacians brings an elegant mean for combining different types of data, and has huge potential for future development.

Biomolecular Topology: Modelling and Analysis

“…Note that the persistent Laplacian provides both topological and spectral information for the characterization of data. Persistent homology based models have been used in molecular data analysis [200,204,205].…”

Persistent topological Laplacian analysis of SARS-CoV-2 variants

“…Inspired by this success, various new TDA methods have been proposed [23,17,1,24]. Recently, an elegant theory, persistent sheaf Laplacian, was proposed to embed heterogeneous information, such as geometry and charge, in topological analysis [45]. Persistent sheaf Laplacian can be regarded as a generation of cellular sheaves [18].…”

“…Both persistent Laplacian and persistent sheaf Laplacian belong to a class of persistent topological Laplacians (PTLs) [43]. PTLs are a family of multiscale topological spectral methods, including continuous (evolutionary) Hodge Laplacians defined on manifolds [12] and all other discrete multiscale topological Laplacians, namely, persistent sheaf Laplacians [45], persistent spectral graphs [39], persistent path Laplacians [40], persistent topological hypergraph Laplacians [6], persistent hyperdigraph Laplacians [6], etc. PTLs contain multicalse topological invariants in their harmonic spectra and homotopic shape evolution in the multiscale non-harmonic spectra.…”

Persistent Path Laplacian

“…That is to say, each point does not carry any labeled information such as the type, mass, color, etc. Therefore, an extension of PSG, called persistent sheaf Laplacian (PSL), was proposed to generalize cellular sheaves [21,22] for the multiscale analysis of point cloud data with attached labeled information [23]. PSL is also a topological Laplacian that carries topological information in its null space but tracks homotopic shape evolution during filtration.…”