Signal processing over graphs has recently attracted significant attention for dealing with the 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 article, we propose a new framework of hypergraph signal processing (HGSP) based on the tensor representation to generalize the traditional graph signal processing (GSP) to tackle highorder interactions. We introduce the core concepts of HGSP and define the hypergraph Fourier space. We then study the spectrum properties of hypergraph Fourier transform (HGFT) 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.

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