Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

Researchers have introduced a novel convolutional learning framework named HilbNets, designed to handle sophisticated, infinite-dimensional signals such as time series and probability distributions defined over irregular domains. Addressing the lack of unified learning theory for these complex settings, the study utilizes the connection Laplacian associated with a Hilbert bundle as a convolutional operator. The framework ensures implementability through a two-stage sampling procedure. First, it demonstrates that sampling the manifold induces a Hilbert Cellular Sheaf, proving its sheaf Laplacian converges to the underlying connection Laplacian as sampling density increases. This generalizes the foundational Belkin and Niyogi convergence result to infinite-dimensional bundles. Second, the authors prove that discretized HilbNets converge to continuous architectures and remain transferable across different samplings, ensuring learning consistency. Validated on both synthetic and real-world tasks, this work significantly broadens geometric learning by extending classical Laplacian-based frameworks to scenarios where signals at each point reside in their own Hilbert space, offering a robust theoretical foundation for modern deep learning applications involving complex data structures.