SheafLapNet: Topology-Aware Neural Architecture

• SheafLapNet is a neural architecture that leverages cellular sheaves and sheaf Laplacians to enable topology- and geometry-aware message passing on graphs, manifolds, simplicial complexes, and hypergraphs.
• It generalizes traditional Laplacian-based diffusion by replacing standard operators with sheaf Laplacians, resulting in localized, anisotropic, and physically interpretable signal propagation.
• The framework integrates persistent topological features and provable convergence guarantees, demonstrating improved performance in tasks such as node classification, protein modeling, and manifold denoising.

SheafLapNet is a class of neural architectures that generalize classical graph neural networks (GNNs) by leveraging the algebraic-topological framework of cellular sheaves and their associated sheaf Laplacians to enable expressive, topology- and geometry-aware message passing on graphs, manifolds, simplicial complexes, and hypergraphs. The defining characteristic of SheafLapNet is the replacement of standard Laplacian-based diffusion with sheaf Laplacian operators, thereby enabling localized, anisotropic, and physically interpretable propagation of signals that capture heterogeneous or structured relational data.

1. Mathematical Foundations: Sheaf Laplacians and Connection Laplacians

At the core of SheafLapNet is the sheaf Laplacian, an operator constructed from a cellular sheaf ℱ on a combinatorial object such as a graph, simplicial complex, or hypergraph. For a graph $G=(V,E)$, a cellular sheaf assigns to every vertex $v$ and edge $e$ a finite-dimensional real vector space (“stalk”), together with restriction maps $$\mathcal{F}_{v\to e}:\mathcal{F}(v)\to\mathcal{F}(e)$$. The sheaf Laplacian $L_\mathcal{F} = \delta^T\delta$ is defined via the sheaf coboundary operator $\delta$ acting on 0-cochains, encoding how local feature assignments fail to agree globally with respect to these restriction maps (Hansen et al., 2020).

When all stalks and restriction maps are trivial (i.e., scalar and the identity), the sheaf Laplacian reduces to the classical graph Laplacian. When the restriction maps are orthogonal transformations, the sheaf Laplacian becomes a discrete analog of the Connection Laplacian, which in differential geometry governs the heat flow of tangent vector fields on manifolds (Battiloro et al., 2022, Barbero et al., 2022, Battiloro et al., 2023). This generalized Laplacian supports message passing between feature spaces of arbitrary dimension and relation structure, supporting richer relational modeling, particularly in settings with signed, asymmetric, or directional data.

2. Continuous-To-Discrete Principle: From Manifolds to Cellular Sheaf Neural Networks

The continuous–discrete paradigm in SheafLapNet begins with manifold-based models, such as Tangent Bundle Neural Networks (TNNs), operating on vector fields $F \in \Gamma(T\mathcal{M})$ of a smooth Riemannian manifold $\mathcal{M}$. Continuous convolution is defined spectrally using the Connection Laplacian $\Delta$, leading to filters $h(\Delta)F$ constructed via spectral multipliers on the manifold’s tangent bundle (Battiloro et al., 2022, Battiloro et al., 2023). The network layer applies combinations of these tangent-bundle filters with differential-preserving nonlinearities.

Spatial discretization proceeds by sampling the manifold at $n$ points and locally estimating tangent spaces using PCA. Parallel transport between tangent frames is approximated by optimal orthogonal maps (obtained via SVD), leading to an orthogonal cellular sheaf (an $O(d)$-bundle) over the resulting proximity graph. The corresponding block-structured sheaf Laplacian $\Delta_n$ then governs message passing between local tangent-feature coordinates.

Time discretization approximates the continuous heat diffusion by a finite impulse response (FIR) polynomial in powers of $e^{\Delta_n}$, yielding a stackable, layerwise network. The discrete model, SheafLapNet, thus acts on sheaf-assigned feature tensors via compositions of sheaf Laplacian-powered convolution, nonlinearities, and channel mixing (Battiloro et al., 2022, Battiloro et al., 2023).

A fundamental spectral convergence theorem guarantees that, under suitable sampling and regularity conditions, discrete SheafLapNet converges to its continuous manifold counterpart as $n\to\infty$ in the high-probability sense (Battiloro et al., 2022, Battiloro et al., 2023).

3. Persistent Sheaf Laplacians and Topological Deep Learning

SheafLapNet incorporates persistent sheaf Laplacians (PSL) to capture multi-scale topological features in data. These are defined for filtered simplicial complexes $K\subseteq L$ and sheaves $\mathscr{S}$ over these complexes, leading to the persistent Laplacian

$$\Delta_q^{L,K} = \partial_{q+1}^{L,K} (\partial_{q+1}^{L,K})^* + (\partial_q^K)^*\partial_q^K$$

and its sheaf-theoretic generalization (where $d^q_{\mathscr S,\mathscr T}$ is the sheaf coboundary operator restricted to appropriate subspaces) (Wei et al., 2021, Ren et al., 18 Jan 2026).

The kernel of PSL encodes persistent cohomology classes, while its nonzero spectrum reflects “approximate” cohomological constraints and physically meaningful relations among features (such as in protein modeling, where stalks and restriction maps are parameterized by atomic charges and geometric distances). SheafLapNet can thus learn from sheaf Betti numbers, eigenvalue statistics, and associated invariants computed at multiple filtration scales, supporting physically grounded and interpretable topological deep learning (Ren et al., 18 Jan 2026).

4. Model Architecture: Sheaf-Laplacian Message Passing

A typical SheafLapNet layer implements sheaf-diffusion-based message passing. For $X$ representing concatenated sheaf-assigned features, a generic update is given by

$$H^{(\ell+1)} = \sigma\left( (I - \widetilde{L}_\mathcal{F}) (I \otimes W_1^{(\ell)}) H^{(\ell)} W_2^{(\ell)} \right)$$

where $\widetilde{L}_\mathcal{F}$ is a (possibly normalized) sheaf Laplacian, $W_1^{(\ell)}$ and $W_2^{(\ell)}$ are learnable channel and feature-mixing weights, and $\sigma$ is a pointwise nonlinearity. The structure of $L_\mathcal{F}$ and the restriction maps (often learned via MLPs for flexible relational modeling, or precomputed as orthogonal for geometric regularization) determines the precise form of message passing (Hansen et al., 2020, Barbero et al., 2022, Duta et al., 2023, Choi et al., 9 May 2025).

Alternative forms employ polynomial spectral filters or persistent-homology-driven attention mechanisms based on the spectral or topological features of the underlying sheaf Laplacian (Wei et al., 2021, Cesa et al., 2023). For hypergraphs or higher-order data, SheafLapNet is adapted to operate on symmetric simplicial sets or hypergraph-specific sheaf Laplacians (Duta et al., 2023, Choi et al., 9 May 2025).

In practical implementations, non-linear and multi-modal fusion, as well as variable stalk dimensions (e.g., for multiple data modalities, or element-specific attributes), are supported by block-diagonal or cross-weighted operator constructions. Regularization and computational scalability are addressed by parameter sharing, precomputation, and variant architectures (linear vs. non-linear Laplacians).

5. Empirical and Theoretical Properties

SheafLapNet architectures demonstrate consistent empirical advantages across diverse regimes:

• On manifold-structured data (e.g., tangent field denoising on spheres or tori), SheafLapNet outperforms manifold-CNNs or basic GNNs by leveraging bundle geometry and orthogonal parallel transport, with lower MSE and sharper recovery of vector fields (Battiloro et al., 2022, Battiloro et al., 2023).
• For node classification tasks with strong heterophily or asymmetric/signed edge relations, SheafLapNet delivers 1–6% higher accuracy than graph Laplacian-based models, due to its ability to encode directionality and multi-way feature interactions in restriction maps (Barbero et al., 2022, Hansen et al., 2020).
• In persistent sheaf formulations, SheafLapNet achieves state-of-the-art results on protein stability/regression and mutation solubility classification, with substantial improvements in correlation coefficients and interpretability metrics over persistent homology and previous topological deep learning models (Ren et al., 18 Jan 2026).
• Hypergraph versions of SheafLapNet, using symmetric simplicial sets, overcome adjacency sparsity, orientation ambiguity, and information loss associated with clique expansions, achieving the highest test accuracy across both homophilic and heterophilic hypergraph benchmark datasets (Duta et al., 2023, Choi et al., 9 May 2025).

Provable properties include:

• Standing spectral convergence guarantees to the continuous (manifold) limit under mild sampling and regularity assumptions (Battiloro et al., 2022, Battiloro et al., 2023).
• Strict contraction of sheaf Dirichlet energy per layer for linear Laplacians, and monotonic decrease of total variation for non-linear variants (Duta et al., 2023).
• Structural generalization of Hodge decomposition and robustness of the sheaf Laplacian spectrum to noise and filtration perturbations (Wei et al., 2021).

6. Extensions: Persistent Local Homology, Higher-Order Structures, and Hypergraphs

Recent advances extend SheafLapNet to persistent local homology sheaves, where node feature spaces are computable as local homology groups (often in degree 1 to capture cycles and stratification). These enable the architecture to align message spaces with the actual “shape” of local neighborhoods, particularly in non-manifold or stratified data (Cesa et al., 2023). Persistent variants enable end-to-end differentiability, supporting optimization of both network weights and the underlying topological structures.

Higher-order and hypergraph SheafLapNets are constructed by equipping hyperedges or higher simplices with vector spaces and defining restriction maps according to multi-way or facet relations. The normalized degree-0 (or higher) sheaf Laplacian on induced symmetric simplicial sets yields mathematically consistent extensions that reduce to standard graph/sheaf Laplacian cases and retain the full provenance of higher-order relations (Duta et al., 2023, Choi et al., 9 May 2025).

7. Significance and Implementation Guidelines

SheafLapNet provides a universal framework that unifies geometric, algebraic-topological, and relational deep learning. It generalizes Laplacian-based diffusion to arbitrary local feature spaces and relations, supports multi-scale and heterogeneous topological encoding (persistent cohomology, local/persistent homology, bundle geometry), and is adaptable to arbitrary combinatorial structures including graphs, simplicial complexes, and hypergraphs.

For optimal performance:

• The choice of stalk dimension and structure should align with the intrinsic data modalities (e.g., tangent bundle dimension, physicochemical channels).
• Orthogonality and/or data-driven learning of restriction maps can be selected according to the amount of available structure and computational cost.
• Persistent sheaf Laplacians, multi-scale pooling, or attention mechanisms should be employed for tasks requiring explicit topological invariance or interpretability.
• Standard techniques for regularization and scalability (dropout, layer normalization, weight tying) are directly applicable.

SheafLapNet thus establishes a rigorous, expressive, and geometrically interpretable class of neural architectures, validated both theoretically and across a broad spectrum of application domains (Hansen et al., 2020, Wei et al., 2021, Battiloro et al., 2022, Barbero et al., 2022, Battiloro et al., 2023, Cesa et al., 2023, Duta et al., 2023, Choi et al., 9 May 2025, Ren et al., 18 Jan 2026).