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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

Chinese

我们将Bochner-Weitzenböck分解扩展到细胞切割的Laplacians,将其分成一个积极的半定义术语,Bochner切割的Laplacian,和一个区块圆形术语,Forman曲线。我们展示了Bochner Laplacians形成一个细胞切割的类别,延伸到连接和矩阵重量Laplacians,并将签署的图表概括为$d$尺寸。我们给他们的内核一个全能性特征,证明相关的Cheeger不平等,并附加到它们上一个随机步行,将Gremban扩张概括。对于Forman术语,我们证明了一个分散的Bochner定理,将一个细胞切割的第一个共体学与非负面的曲线联系起来,我们将整个分

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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