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

This paper proposes a unified sheaf signal processing (SSP) framework for networks whose nodes carry heterogeneous local signal spaces. It extends spectral analysis, filtering, and sampling beyond conventional graph signal processing, where signals usually share a common ambient space. The proposed Sheaf Fourier Transform measures inconsistency induced by topology, restriction maps, and local geometry. The work also develops polynomial sheaf filters, joint node-and-component sampling, perfect-recovery conditions for bandlimited sheaf signals, and a greedy sampling algorithm. Representation sheaves support alternative bases, dictionaries, and learned embeddings. The authors report experiments on synthetic, motion-capture, and financial datasets against canonical graph signal processing baselines.