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Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

NeurIPS 2026 · Graph learning › GNN architectures · approach: Theory of graph convolutional networks · 0 citations (OpenAlex)

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Introduces HilbNets, convolutional networks on Hilbert bundles over manifolds, discretized via Hilbert cellular sheaves with proven Laplacian convergence and transferability guarantees, validated on a synthetic transport task and traffic forecasting.

Problem

Convolutional learning for infinite-dimensional signals (e.g., time series, distributions) defined over manifolds lacks a theory; existing Laplacian convergence results only cover finite-dimensional fibers.

Approach

Define convolution via the connection Laplacian of a Hilbert bundle and the Borel functional calculus, then discretize by sampling the manifold into a Hilbert cellular sheaf whose sheaf Laplacian converges to the connection Laplacian, plus fiber projection to finite rank.

Key results

• On METR-LA at horizon 3, HilbNet free O(T) reaches MAE 2.923 vs 3.131 for the MLP fiber baseline and 3.453 for the spatiotemporal graph baseline.
• On the synthetic transport task, the free O(d) class recovers Levi-Civita transports to numerical precision (~1.6e-7 edge-MSE).
• HilbNet circulant uses roughly one tenth of the transport parameters of the free variant while remaining competitive on traffic forecasting.

Surprising findings

• On PEMS-BAY at horizon 3, the spatiotemporal graph baseline achieves the best MAE (1.400) and MAPE, beating HilbNet variants.

Limitations

• Convergence theory assumes closed manifolds, C^3/C^4 smooth sections, and uniform iid sampling, which real sensor networks do not satisfy.
• Sheaf Laplacian scales quadratically in n*d, which may be prohibitive for large graphs without sparsification.
• Empirical validation limited to synthetic and traffic data; distributional/functional data not tested.

Experimental setup

Datasets: METR-LA PEMS-BAY synthetic Gaussian bundle over Sym++(p)

Benchmarks: METR-LA PEMS-BAY

Metrics: MAE RMSE MAPE edge-MSE

Baselines: FC-LSTM STAEformer MLP fiber baseline spatiotemporal graph baseline HilbNet frozen identity (GCN) HilbNet circulant

Builds on: Belkin & Niyogi graph Laplacian convergence sheaf neural networks graph convolutional networks tangent bundle neural networks

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