Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling

2026-08-02

AI Summary

This work addresses the challenge that conventional graph signal processing methods struggle to effectively model node data in heterogeneous networks due to disparities in dimensionality, modality, and geometric structure. To overcome this limitation, the authors propose a unified framework termed Layered Signal Processing (SSP), which characterizes heterogeneous local signal spaces through network layers and the linear mappings between them, thereby generalizing fundamental operations such as spectral analysis, filtering, and sampling. Key contributions include the first formal definition of the Layered Fourier Transform (SFT), whose frequency basis is constructed from topological and restriction mappings; the introduction of representation layers that accommodate diverse bases, dictionaries, or embeddings while preserving spectral properties; and the design of polynomial layered filters along with a joint node-component sampling strategy. Experiments on synthetic, motion capture, and financial datasets demonstrate significant performance gains over classical baselines, and the framework establishes conditions for perfect reconstruction of bandlimited signals.

Abstract

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.

Problem

• heterogeneous signals
• network sheaves
• signal processing
• local signal spaces
• graph signals

Innovation

• Sheaf Signal Processing
• Sheaf Fourier Transform
• Heterogeneous Signal Spaces
• Restriction Maps
• Representation Sheaves