Analysis of Hypergraph Signals via High-Order Total Variation

Beyond pairwise relationships, interactions among groups of agents do exist in many real-world applications, but they are difficult to capture by conventional graph models. Generalized from graphs, hypergraphs have been introduced to describe such high-order group interactions. Inspired by graph signal processing (GSP) theory, an existing hypergraph signal processing (HGSP) method presented a spectral analysis framework relying on the orthogonal CP decomposition of adjacency tensors. However, such decomposition may not exist even for supersymmetric tensors. In this paper, we propose a high-order total variation (HOTV) form of a hypergraph signal (HGS) as its smoothness measure, which is a hyperedge-wise measure aggregating all signal values in each hyperedge instead of a pairwise one in most existing work. Further, we propose an HGS analysis framework based on the Tucker decomposition of the hypergraph Laplacian induced by the aforementioned HOTV. We construct an orthonormal basis from the HOTV, by which a new spectral transformation of the HGS is introduced. Then, we design hypergraph filters in both vertex and spectral domains correspondingly. Finally, we illustrate the advantages of the proposed framework by applications in label learning.