# Introducing Hypergraph Signal Processing: Theoretical Foundation and Practical Applications

## TL;DR

This paper introduces hypergraph signal processing (HGSP), a tensor-based framework that generalizes graph signal processing to model high-order relationships, with theoretical foundations and practical applications demonstrating improved performance.

## Contribution

The paper develops the theoretical foundation of HGSP, including hypergraph Fourier space, spectrum properties, sampling theory, and filter design, extending GSP to high-order data interactions.

## Key findings

- HGSP outperforms traditional methods in experimental tests.
- Hypergraph Fourier transform captures high-order relationships effectively.
- The framework enables advanced signal processing in IoT and complex data scenarios.

## Abstract

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.