Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

Modern deep learning increasingly contends with signals that are natively infinite-dimensional and supported on irregular domains, yet a unified learning theory for such settings has been lacking. We introduce a convolutional framework for possibly infinite-dimensional signals on a manifold by using the connection Laplacian of a Hilbert bundle as the convolutional operator, yielding filters and neural networks we call HilbNets. To make this implementable, we develop a two-stage sampling procedure: first, sampling the manifold induces a Hilbert Cellular Sheaf whose sheaf Laplacian converges in probability to the underlying connection Laplacian, generalizing the Belkin & Niyogi graph-to-manifold Laplacian convergence to the infinite-dimensional bundle setting; second, we discretize the signals and prove that the resulting HilbNets converge to their continuous counterparts and are transferable across samplings of the same bundle, ensuring consistency for learning. We validate the framework on synthetic and real-world tasks. Overall, our results broaden geometric learning by lifting classical Laplacian-based frameworks to settings where the signal at each point lives in its own Hilbert space.