# Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

arXiv: https://arxiv.org/abs/2605.06395 · HTML: https://mlsanity.com/paper/2605.06395/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 112 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 21 of 21)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Equivariant Contrastive Learning](https://mlsanity.com/paper/2111.00899/) (2021)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [Convolutional Neural Networks on Manifolds: From Graphs and Back](https://mlsanity.com/paper/2210.00376/) (2022)
- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Stability to Deformations of Manifold Filters and Manifold Neural Networks](https://mlsanity.com/paper/2106.03725/) (2021)
- [Parallel Transport Convolution: A New Tool for Convolutional Neural Networks on Manifolds](https://mlsanity.com/paper/1805.07857/) (2018)
- [Semantic Tube Prediction: Beating LLM Data Efficiency with JEPA](https://mlsanity.com/paper/2602.22617/) (2026)
- [Distributed Multi-agent Coordination over Cellular Sheaves](https://mlsanity.com/paper/2504.02049/) (2025)
- [Spherical Analysis of Learning Nonlinear Functionals](https://mlsanity.com/paper/2410.01047/) (2024)
- [Sheaves Reloaded: A Directional Awakening](https://mlsanity.com/paper/2506.02842/) (2025)
- [Learning the Structure of Connection Graphs](https://mlsanity.com/paper/2510.11245/) (2025)
- [Learning Network Sheaves for AI-native Semantic Communication](https://mlsanity.com/paper/2512.03248/) (2025)
- [Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning](https://mlsanity.com/paper/2604.20308/) (2026)
- [Learning from Frustration: Torsor CNNs on Graphs](https://mlsanity.com/paper/2510.23288/) (2025)
- [Graph Distribution-valued Signals: A Wasserstein Space Perspective](https://mlsanity.com/paper/2509.25802/) (2025)
- [Change Point Detection for Functional Autoregressive Processes on the Sphere](https://mlsanity.com/paper/2512.03255/) (2025)
- [Size-Transferability of Graph Transformers with Convolutional Positional Encodings](https://mlsanity.com/paper/2602.15239/) (2026)

## Built on by (most cited first, 1 of 1)

- [Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling](https://mlsanity.com/paper/2608.01318/) (2026)

## Abstract (arXiv)

Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as textitHilbNets. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin & Niyogi citeBELKIN20081289 convergence result for the graph Laplacian to the manifold Laplacian, a theoretical cornerstone of geometric learning methods. Second, we discretize the signals and prove that the discretized (implementable) HilbNets converge to the underlying continuous architectures and are transferable across different samplings of the same bundle, providing consistency for learning. Finally, we validate our framework on synthetic and real-world tasks. Overall, our results broaden the scope of geometric learning as a whole by lifting classical Laplacian-based frameworks to settings where the signal at each point lives in its own Hilbert space.