A Bochner Weitzenb\"ock Decomposition for Cellular Sheaf Laplacians, Curvature and Neural Sheaf Diffusion

Abstract:We extend the Bochner-Weitzenböck decomposition to cellular sheaf Laplacians, splitting it into a positive semidefinite term, the Bochner sheaf Laplacian, and a block diagonal term, the Forman curvature. We show that the Bochner Laplacians form a class of cellular sheaves that extends both connection and matrix-weighted Laplacians, and generalizes signed graphs to $d$ dimensions. We give a holonomy characterization of their kernel, prove the related Cheeger inequalities, and attach to them a random walk that generalizes the Gremban expansion. For the Forman term, we prove a discrete Bochner theorem, which bounds the first cohomology of a cellular sheaf with nonnegative curvature, and we read the whole decomposition as an opinion-dynamics model. We then apply this machinery to Sheaf Neural Networks, whose learnt Laplacian is otherwise unobserved. The Bochner interpretation yields a node-level conviction score that acts as an operational importance measure under frozen-model interventions, while the Forman profile provides a local descriptor of the learnt operator.

Computational Geometry (cs.CG); Machine Learning (cs.LG)

arXiv:2610.28455 [cs.CG]

(or arXiv:2610.28455v1 [cs.CG] for this version)

https://doi.org/10.48550/arXiv.2610.28455 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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