Implicit Sheaf Operators for Effective and Efficient Cellular Sheaf Learning

Gabriele Onorato; Dario Loi; Cristian Apostol; Nathaniel D. Bastian; Francesco Restuccia

Implicit Sheaf Operators for Effective and Efficient Cellular Sheaf Learning

Gabriele Onorato
Dario Loi
Cristian Apostol
Nathaniel D. Bastian
Francesco Restuccia

Preprint, August 2026

Left, a 3 by 3 block sheaf Laplacian with d by d blocks multiplied by a stacked feature matrix X. Right, a triangle of nodes v, u, w with edges e1 and e2, where dashed arrows bring restriction-map terms from u and w, and a self term, into node v. — Figure 2.(Left) The Sheaf Laplacian (SL) $L_{\mathcal{F}} \in \mathbb{R}^{nd \times nd}$ is explicitly computed before multiplying it with the sheaf node features $\mathbf{X}$ . **(Right)** The proposed ISOs reformulate the computation as a purely local process. By leveraging the topological features of cellular sheaves, ISO computes the exact action of $L_{\mathcal{F}}\mathbf{X}$ independently at each single node $v$ , bypassing the global matrix completely.

In short Implicit Sheaf Operators compute sheaf diffusion as local operator-vector products instead of building the sheaf Laplacian, with up to 4.87× lower latency and 3.92× less peak GPU memory.

Abstract

While Sheaf Neural Networks (SNNs) enrich $n$ -node $m$ -edge diffusion of $f$ -dimensional node features with $d$ -dimensional node and edge stalks, they require computing a sparse $nd \times nd$ sheaf Laplacian, whose computation is prohibitive on edge devices. This paper addresses this issue with Implicit Sheaf Operators (ISOs), a type of sheaf topological operators that leverage the topological structure of cellular sheaves to decompose the global diffusion process into local operator-vector products. We mathematically prove that ISO eliminates the $O(md^2 \log(md^2))$ sparse-coalescing term and reduces the space complexity by $O(\min(d, f))$ with respect to traditional Neural Sheaf Diffusion (NSD). Our experiments indicate that ISO-based operators achieve latency reduction of up to 4.87× and save up to 3.92× peak GPU memory. On real graph classification data, ISO uses up to 1.24× less energy and 3.08× less on-device memory as measured on a Jetson Orin Nano. Finally, to bring ISO to practice, we introduce Sheaf Convolutional Networks (SCNs), an ISO-based SNN attaining up to 3.93% higher accuracy compared to NSD-based SNNs on node classification benchmarks.

Cite

BibTeX

@misc{onorato2026implicit,
  title  = {Implicit Sheaf Operators for Effective and Efficient Cellular Sheaf Learning},
  author = {Gabriele Onorato and Dario Loi and Cristian Apostol and Nathaniel D. Bastian and Francesco Restuccia},
  year   = {2026},
  note   = {Preprint},
  url    = {https://mentis.info/wp-content/uploads/2026/08/SheafOperators.pdf}
}