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A Bochner Weitzenböck Decomposition for Cellular Sheaf Laplacians, Curvature and Neural Sheaf Diffusion

Francesco Maria Guadagnuolo, Alessio Borgi, Dorina Thanou, Pietro Liò · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28455 · cited by 0 works (OpenAlex) · open-access PDF

German

Wir erweitern die Bochner-Weitzenböck-Auflösung auf zelluläre Scheiben Laplacians, teilen sie in einen positiven halbdefinierten Begriff, den Bochner-Schieben Laplacian, und einen Block-Diagonalen Begriff, die Forman-Krümmung. Wir zeigen, dass die Bochner-Laplacians eine Klasse von Zellschichten bilden, die sowohl die Verbindung als auch die matrixgewichtete Laplacians erweitert und unterzeichneten Grafen zu $d$-Dimensionen verallgemeinert. Wir geben eine holonomische Charakterisierung ihres Kernels, beweisen die damit verbundenen Cheeger-Ungleichheiten und fügen ihnen einen zufälligen Spaziergang hinzu, der die Gremban-Erweiterung verallgemeinert

Machine translation (Meta M2M100); the English original follows.

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.

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Topics: Mathematics, Sheaf, Sheaf cohomology, Pure mathematics, Measure (data warehouse), Laplace operator

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