[Paper Review] Persistent sheaf Laplacians

Xiaoqi Wei, Guo‐Wei Wei|arXiv (Cornell University)|Dec 20, 2021

Topological and Geometric Data Analysis8 citations

TL;DR

This paper introduces persistent sheaf Laplacians (PSLs), a novel framework that extends persistent Laplacians to cellular sheaves, enabling multiscale spectral analysis of point clouds with both geometric and non-geometric (e.g., atomic charges, node properties) data. By constructing sheaves on labeled simplicial complexes and applying filtration, PSLs capture topological invariants and shape evolution while encoding localized physical properties, offering a unified tool for data fusion in complex systems like molecules and networks.

ABSTRACT

Recently various types of topological Laplacians have been studied from the perspective of data analysis. 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 cellular sheaves, this work develops the theory of persistent sheaf Laplacians for cellular sheaves, and describes how to construct sheaves for a point cloud where each point is associated with a quantity that can be devised to embed physical properties. As a result, the spectra of persistent sheaf Laplacians encode both geometrical and non-geometrical information of the given point cloud. The theory of persistent sheaf Laplacians is an elegant method for fusing different types of data and has huge potential for future development.

Motivation & Objective

• To overcome the limitation of persistent homology and persistent Laplacians in capturing non-geometric, local data properties such as atomic charges or node degrees.
• To develop a theoretical framework that extends persistent Laplacians to cellular sheaves, enabling spectral analysis of data with both topological and physical structure.
• To demonstrate how sheaf cohomology and sheaf Laplacians can be made persistent through filtration, preserving topological invariants while capturing shape evolution.
• To enable data fusion by embedding physical laws and mathematical rules into sheaf assignments on point clouds, particularly for molecular and network data.
• To lay the foundation for future extensions, including persistent sheaf Dirac theory and evolutionary sheaf analysis on manifolds.

Proposed method

• Construct a labeled simplicial complex from a point cloud, assigning vector spaces (e.g., R) to vertices based on local data such as atomic charges.
• Define a cellular sheaf by assigning linear maps between vector spaces for face relations, forming a cochain complex for sheaf cohomology.
• Introduce a filtration of simplicial complexes to generate a family of nested complexes, enabling persistent sheaf cohomology and persistent sheaf Laplacians.
• Formulate the q-th persistent sheaf Laplacian as a matrix operator on the sheaf cochain complex, generalizing the persistent combinatorial Laplacian to sheaf settings.
• Compute the spectra of persistent sheaf Laplacians across filtration levels, capturing both harmonic (topological) and non-harmonic (geometric/physical) spectral components.
• Apply the method to synthetic shapes and real molecular systems (e.g., cucurbit[8]uril, bacteriocin AS-48), using partial charges and force field data to define sheaf structures.

Experimental results

Research questions

• RQ1 How can persistent Laplacians be generalized to handle non-geometric, localized data such as atomic charges or node properties in a topologically meaningful way?
• RQ2 Can sheaf cohomology and sheaf Laplacians be made persistent through filtration to retain topological invariants while capturing shape evolution?
• RQ3 What is the spectral behavior of persistent sheaf Laplacians on point clouds with varying geometric and physical structures?
• RQ4 How do the spectra of persistent sheaf Laplacians compare to those of constant sheaves or standard persistent Laplacians in capturing data complexity?
• RQ5 Can persistent sheaf Laplacians effectively fuse geometric and non-geometric data for applications in molecular modeling and network science?

Key findings

• Persistent sheaf Laplacians successfully encode both geometric structure and non-geometric data (e.g., atomic charges) in the spectra, enabling multiscale analysis beyond persistent homology.
• For trapezoid and square shapes, the spectra of sheaf Laplacians with non-uniform charges show distinct patterns compared to constant sheaves, reflecting localized physical properties.
• In cucurbit[8]uril, the spectral results at p=0.2 reveal complex, non-trivial patterns that reflect the molecule’s charge distribution and topology, even when geometric complexity is high.
• For bacteriocin AS-48, the spectra at p=0.4 show clear differences from the p=0 case, indicating sensitivity to shape evolution and localized charge distributions in protein structures.
• The method demonstrates robustness in capturing subtle spectral changes due to both topological and physical variations, suggesting utility in machine learning and data fusion.
• The framework provides a foundation for future development, including fast implementations, physical law integration, and persistent sheaf Dirac theory.