# Introducing Hypergraph Signal Processing: Theoretical Foundation and Practical Applications

arXiv: https://arxiv.org/abs/1907.09203 · HTML: https://mlsanity.com/paper/1907.09203/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2019.

## Built on by (most cited first, 20 of 20)

- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Equivariant Hypergraph Diffusion Neural Operators](https://mlsanity.com/paper/2207.06680/) (2022)
- [Finite Impulse Response Filters for Simplicial Complexes](https://mlsanity.com/paper/2103.12587/) (2021)
- [Causal Fourier Analysis on Directed Acyclic Graphs and Posets](https://mlsanity.com/paper/2209.07970/) (2022)
- [Adaptive Neural Message Passing for Inductive Learning on Hypergraphs](https://mlsanity.com/paper/2109.10683/) (2021)
- [Learning Hypergraphs From Signals With Dual Smoothness Prior](https://mlsanity.com/paper/2211.01717/) (2022)
- [Training-Free Message Passing for Learning on Hypergraphs](https://mlsanity.com/paper/2402.05569/) (2024)
- [Hypergraph-MLP: Learning on Hypergraphs without Message Passing](https://mlsanity.com/paper/2312.09778/) (2023)
- [Towards Multi-agent Reinforcement Learning based Traffic Signal Control through Spatio-temporal Hypergraphs](https://mlsanity.com/paper/2404.11014/) (2024)
- [Hyperedge Representations with Hypergraph Wavelets: Applications to Spatial Transcriptomics](https://mlsanity.com/paper/2409.09469/) (2024)
- [An Efficient Hypergraph Approach to Robust Point Cloud Resampling](https://mlsanity.com/paper/2103.06999/) (2021)
- [SPHINX: Structural Prediction using Hypergraph Inference Network](https://mlsanity.com/paper/2410.03208/) (2024)
- [Scalable Hypergraph Structure Learning with Diverse Smoothness Priors](https://mlsanity.com/paper/2504.03583/) (2025)
- [Regularized Recovery by Multi-order Partial Hypergraph Total Variation](https://mlsanity.com/paper/2102.09771/) (2021)
- [Tensor Laplacian Regularized Low-Rank Representation for Non-uniformly Distributed Data Subspace Clustering](https://mlsanity.com/paper/2103.04064/) (2021)
- [Spectral Convergence of Complexon Shift Operators](https://mlsanity.com/paper/2309.07169/) (2023)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [A Framework for Directed Hypergraph Signal Processing via tensor t-SVD](https://mlsanity.com/paper/2606.25112/) (2026)
- [Topological Signal Processing With Unoriented Operators](https://mlsanity.com/paper/2609.25310/) (2026)

## Abstract (arXiv)

Signal processing over graphs has recently attracted significant attentions for dealing with structured data. Normal graphs, however, only model pairwise relationships between nodes and are not effective in representing and capturing some high-order relationships of data samples, which are common in many applications such as Internet of Things (IoT). In this work, we propose a new framework of hypergraph signal processing (HGSP) based on tensor representation to generalize the traditional graph signal processing (GSP) to tackle high-order interactions. We introduce the core concepts of HGSP and define the hypergraph Fourier space. We then study the spectrum properties of hypergraph Fourier transform and explain its connection to mainstream digital signal processing. We derive the novel hypergraph sampling theory and present the fundamentals of hypergraph filter design based on the tensor framework. We present HGSP-based methods for several signal processing and data analysis applications. Our experimental results demonstrate significant performance improvement using our HGSP framework over some traditional signal processing solutions.