An introduction to hypergraph signal processing

Developing tools to analyze signals defined over a graph is a research area that is attracting a significant amount of contributions because of its many applications. However, a graph representation does not capture the overall information about the data, as it implicitly takes into account only pairwise relations. The goal of this paper is to extend signal processing tools to signals defined over hypergraphs, which represent a formal framework to describe multi-way relations among the data. First, we suggest alternative ways to introduce a Fourier Transform (FT) for signals defined over hypergraphs and, in particular, for simplicial complexes. Then, building on the notion of Fourier Transform, we derive a sampling theorem aimed at identifying the minimum number of samples necessary to encode all information about band-limited hypergraph signals.