arXiv cs.AI· Kartik Tandon, Julian Gould, Tanishq Bhatia, Francesca Dominici, Alejandro Ribeiro, Claudio Battiloro·· 11 小时前AI 评分24
HilbNets：用 Hilbert 丛与胞腔层实现一致的几何深度学习
Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves
AI 导读
研究者提出基于 Hilbert 丛联络 Laplacian 的卷积学习框架 HilbNets，可处理流形上可能无限维的信号（如时间序列、概率分布、算子）。该工作证明采样流形会诱导 Hilbert 胞腔层，其层 Laplacian 随采样密度增加依概率收敛到联络 Laplacian，并将 Belkin & Niyogi 的图 Laplacian 收敛结果推广到无限维丛设定。
正文 · 原文
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.
| Comments: | NeurIPS 2026 Spotlight. 52 pages, 3 figures, 5 tables |
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Signal Processing (eess.SP) |
| Cite as: | arXiv:2605.06395 [cs.LG] |
| (or arXiv:2605.06395v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.06395 arXiv-issued DOI via DataCite |
Submission history
From: Claudio Battiloro Dr [view email]
[v1]
Thu, 7 May 2026 15:08:58 UTC (4,665 KB)
[v2]
Tue, 19 May 2026 20:23:58 UTC (4,665 KB)
[v3]
Wed, 4 Nov 2026 15:41:49 UTC (4,663 KB)
来源：arXiv cs.AI · arxiv.org