Toward a spectral theory of cellular sheaves

Abstract

This paper outlines a program in what one might call spectral sheaf theory -an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes results on eigenvalue interlacing, complex sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.

The diffusion geometry of fibre bundles: Horizontal diffusion maps

“…Similar consideration motivated topological data analysts to propose sheaves as data models; see e.g. [HG18] and the references therein. Even in cases in which the pairwise correspondences can be modeled as group elements, the group can be too large to manipulate efficiently, such as Lie groups of diffeomorphisms or isometries commonly encountered in non-isometric collection shape analysis [BBK08, HZG + 12, HG13, LZ17].…”

Persistent sheaf Laplacians

“…We assume that the reader is familiar with notions of simplicial homology. Most researchers define cellular sheaves on regular cell complexes [8,11,16,29]. For the sake of simplicity, we only discuss celluar sheaves on simplical complexes, as they are indeed regular cell complexes.…”

“…Hansen and Ghrist [16] apply this construction to sheaf cochain complexes and the resulting combinatorial Laplacian is called the sheaf Laplacian. If every stalk of a cellular sheaf S is a finite dimensional inner product space over R or C, we can equip an inner product on every sheaf cochain group C q (X; S ) by letting S (σ) and S (σ ) orthogonal to each other if σ = σ .…”

“…The notion of a cellular sheaf was first introduced by A. D. Shepard in his Ph.D. thesis in 1985 [31]. In the last decade, R. Ghrist and coworkers and many other researchers have revived and expanded the cellular sheaf theory with applications in science and engineering [8,12,16,29]. Tools from sheaf theory have also been developed to study persistence modules [1,2,8,15].…”

“…Therefore, one can define sheaf cohomology, which reflects the property of the sheaf. Spectral sheaf theory extends spectral graph theory to cellular sheaves, leading to sheaf Laplacians [16].…”