Spectral Sheaf Theory
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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[1]. This approach is referred to as Spectral Sheaf Theory.

# Motivation

In the realm of spectral graph theory, combinatorial graphs are enriched with algebraic structures represented as square matrices, notably degree and adjacency matrices, Laplacian matrices, and their weighted or normalized counterparts. The matrix size typically corresponds to the vertex set, with its structure encapsulating data from the edges. Spectral graph theory, with its profound implications, underpins diverse fields including data analysis, theoretical computer science, probability, control theory, numerical linear algebra, coding theory, and graph theory itself.

A significant portion of this theory emphasizes the Laplacian matrix, capitalizing on its multifaceted interpretations in discrete contexts. Notably, the zero eigenvalue multiplicity of the graph Laplacian indicates the graph's connected components, while the smallest nonzero eigenvalue in a connected graph signifies its near-disconnectivity, showcasing topological characteristics.

Laplacians also play a central role in Hodge theory, a segment of algebraic and differential geometry. Hodge theory employs Laplacians on Riemannian manifolds to delineate global attributes. A foundational result reveals the cohomology of the manifold as the Laplacian's kernel on differential forms. Specifically, the kernel's dimension on 0-forms equates to the rank of $H^0$, the 0-th cohomology group, which corresponds to the number of connected components, thus drawing parallels between Hodge theory and elements of spectral graph theory.

Moving from Riemannian manifolds to weighted cell complexes, Hodge theory's principles remain intact. The classical boundary operator for a cell complex, combined with its formal adjoint, results in a graph Laplacian extension that operates on higher-dimensional objects (cellular cochains, as opposed to differential
forms). This discrete Laplacian's kernel mirrors the cellular cohomology of the complex, extending the graph Laplacian's connectivity detection. Consequently, the discrete Laplacian's spectral theory provides insights into the algebraic-topological aspects of advanced-dimensional complexes, a topic of contemporary exploration.

# Summary

What follows is a condensed list of results and possible applications[1]:

• Harmonic extension: for a pair $(X, A)$ where $A \subset X$, given $H^k(X, A; \mathcal{F}) = 0$, there exists unique harmonic extension of cochains on $A$ to $X$.
• Maximum modulus principle: holds for $\mathrm{O}(n)$-bundles (the $\mathrm{O}(n)$-bundles are discussed further, physical interpretation of the maximum modulus is that dispersion of free parameters in $\mathrm{O}(n)$-valued stalks is asymptotic).
• Effective resistance: a pair of homologous cycles has effective resistance as defined via the minimal-norm bounding chain. (Computable with NP-complete complexity in integer coefficients). Effective resistance acts as a probability on graph cycles, thus allowing for random sampling to compress a sheaf with control of the eigenspectrum of the sheaf Laplacian.
• Sheaf Approximation: while effective resistance computations are NP-complete, using sparse approximations permits further compression and reduced data transfer (this is similar to latent spaces, and is a promising area of further inquiry).
• Expander Sheaves: $\eta$-expander sheaf is a $k$-regular sheaf whose adjacency matrix has $\eta$-bounded spectrum.
• Expander Mixing Lemma: we can compute an expected trace of edges between subsets of a vertex set of a regular sheaf.
 (edit) Topics in Topological Data Analysis
Bibliography
1. Hansen, J., Ghrist, R., (2019), "Toward a spectral theory of cellular sheaves." J Appl. and Comput. Topology 3, 315–358 (doi), (arXiv:1808.01513v2)
page revision: 12, last edited: 22 Aug 2023 05:59