Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

2026-05-07

AI Summary

This work addresses the lack of a unified theoretical framework in geometric deep learning for infinite-dimensional signals—such as time series, probability distributions, or operators—defined on irregular domains. The authors propose a continuous convolution framework grounded in Hilbert bundles and connection Laplacians, which yields computable HilbNets through a two-stage sampling procedure and introduces Hilbert cellular layers to approximate continuous geometric signals. For the first time, convergence theory for graph Laplacians is extended to the infinite-dimensional Hilbert bundle setting, establishing a unified continuous–discrete learning architecture that is consistent and transferable across samplings. Theoretically, discrete HilbNets are shown to converge to their continuous counterparts and exhibit cross-sampling generalization. Empirical validation on both synthetic and real-world tasks demonstrates the effectiveness of the proposed approach.

Abstract

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 \textit{HilbNets}. 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 \cite{BELKIN20081289} 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.

Problem

• Geometric Deep Learning
• Infinite-dimensional Signals
• Irregular Domains
• Hilbert Bundles
• Learning Theory

Innovation

• Hilbert bundles
• Cellular sheaves
• Connection Laplacian
• Geometric deep learning
• Infinite-dimensional signals