基于层论的图上信号处理：谱理论、滤波与采样

中文摘要 AI 辅助

现代传感、通信与学习系统会生成异构网络信号，其中局部数据在维度、模态和几何结构上存在差异。处理此类数据需要一个数学框架，能够同时对异构局部信号空间及其间的变换进行建模。网络层（Network sheaves）提供了这样的框架，它将局部向量空间与网络实体关联，将线性限制映射与实体间的交互关联。本文首次在网络层上开发了统一的层信号处理（Sheaf Signal Processing, SSP）框架，将信号处理的基本操作，即谱分析、滤波和采样，扩展到异构局部空间。与图信号处理和拓扑信号处理中信号建模在公共向量空间不同，SSP联合建模异构局部信号空间以及相邻空间间通过限制映射实现的线性变换。本文定义了层傅里叶变换（Sheaf Fourier Transform, SFT），其频率量化由网络拓扑、限制映射和局部几何结构引起的信号不一致性。基于该表示，本文开发了多项式层滤波器，并将采样形式化为网络节点和节点内分量的联合选择；推导了带限层信号的完美恢复条件，提出了一种贪心采样集设计算法。为纳入依赖应用的信号模型，包括不同基、字典和学习嵌入，本文引入表示层（Representation sheaves），并表征了保留谱性质且保证跨表示互操作性的自然变换。在合成数据集、动作捕捉数据集和金融数据集上的实验验证了所提框架，且其性能相比经典图信号处理基线有持续提升。

英文摘要

Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure. Processing such data requires a mathematical framework capable of simultaneously modeling heterogeneous local signal spaces and the transformations relating them. Network sheaves provide such a framework by associating local vector spaces with network entities and linear restriction maps with their interactions. This is the first paper to develop a unified sheaf signal processing (SSP) framework on network sheaves, extending the fundamental operations of signal processing, namely spectral analysis, filtering, and sampling, to heterogeneous local spaces. Unlike graph and topological signal processing, where signals are modeled over a common vector space, SSP jointly models heterogeneous local signal spaces and the linear transformations relating neighboring spaces through restriction maps. We define the Sheaf Fourier Transform (SFT), whose frequencies quantify signal inconsistency induced by the network topology, the restriction maps, and the local geometry. Building on this representation, we develop polynomial sheaf filters and formulate sampling as the joint selection of network nodes and intra-node components. We derive perfect recovery conditions for bandlimited sheaf signals and propose a greedy sampling-set design algorithm. To incorporate application-dependent signal models, including different bases, dictionaries, and learned embeddings, we introduce representation sheaves and characterize the natural transformations that preserve spectral properties and guarantee interoperability across representations. Experiments on synthetic, motion-capture, and financial datasets validate the proposed framework and demonstrate consistent improvements over canonical graph signal processing baselines.

发表机构

• Sapienza University of Rome（罗马第一大学）
• National Inter-University Consortium for Telecommunications (CNIT)（国家大学间电信联盟（CNIT））

机构由 AI 辅助整理，请以论文原文为准。