Vertex–frequency hypergraph signal processing: Analytic tools and applications

Hypergraph signal processing (HGSP) has attracted the attention of the academic community due to its ability to deal with higher-order interactions. Recently, the Fourier transform gained some versions in this scenario. In a previous work, we introduced a Fourier transform, here simply called the hypergraph Fourier transform (HGFT), which allows us to generalize the windowed Fourier transform to hypergraphs. In this work, we demonstrate how other vertex–frequency analysis tools can be extended to hypergraphs using our HGFT, such as the localized Fourier transform, the spectral wavelet transform, the vertex–frequency energy distribution, the Tikhonov regularization, and the regularization centrality. Several examples using path, squid, and random geometric hypergraphs illustrate the applicability of the proposed methods. Furthermore, some potential applications of these methods are presented, such as semi-supervised classification.