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

arXiv: https://arxiv.org/abs/2608.01318 · HTML: https://mlsanity.com/paper/2608.01318/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 40 references that are in the map.
- First posted on arXiv 2 Aug 2026.

## Builds on (most cited first, 23 of 23)

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Goal-Oriented and Semantic Communication in 6G AI-Native Networks: The 6G-GOALS Approach](https://mlsanity.com/paper/2402.07573/) (2024)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [A Time-Vertex Signal Processing Framework](https://mlsanity.com/paper/1705.02307/) (2017)
- [Simplicial Convolutional Filters](https://mlsanity.com/paper/2201.11720/) (2022)
- [Vector-Valued Graph Trend Filtering with Non-Convex Penalties](https://mlsanity.com/paper/1905.12692/) (2019)
- [Multi-way Graph Signal Processing on Tensors: Integrative analysis of irregular geometries](https://mlsanity.com/paper/2007.00041/) (2020)
- [Topological Signal Processing and Learning: Recent Advances and Future Challenges](https://mlsanity.com/paper/2412.01576/) (2024)
- [Sheaf theory: from deep geometry to deep learning](https://mlsanity.com/paper/2502.15476/) (2025)
- [Adaptive Graph Signal Processing: Algorithms and Optimal Sampling Strategies](https://mlsanity.com/paper/1709.03726/) (2017)
- [Sampling and Recovery of Graph Signals](https://mlsanity.com/paper/1712.09310/) (2017)
- [Forecasting Multi-Dimensional Processes over Graphs](https://mlsanity.com/paper/2004.08260/) (2020)
- [Topological Dictionary Learning](https://mlsanity.com/paper/2503.11470/) (2025)
- [Networks of Causal Abstractions: A Sheaf-theoretic Framework](https://mlsanity.com/paper/2509.25236/) (2025)
- [Learning Network Sheaves for AI-native Semantic Communication](https://mlsanity.com/paper/2512.03248/) (2025)
- [Learning Consistent Causal Abstraction Networks](https://mlsanity.com/paper/2602.02623/) (2026)
- [Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves](https://mlsanity.com/paper/2605.06395/) (2026)

## Abstract (arXiv)

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.