Regularized Recovery by Multi-Order Partial Hypergraph Total Variation

Abstract

Capturing complex high-order interactions among data is an important task in many scenarios. A common way to model high-order interactions is to use hypergraphs whose topology can be mathematically represented by tensors. Existing methods use a fixed-order tensor to describe the topology of the whole hypergraph, which ignores the divergence of differentorder interactions. In this work, we take this divergence into consideration, and propose a multi-order hypergraph Laplacian and the corresponding total variation. Taking this total variation as a regularization term, we can utilize the topology information contained by it to smooth the hypergraph signal. This can help distinguish different-order interactions and represent high-order interactions accurately.

Analysis of Hypergraph Signals via High-Order Total Variation

“…In our work, we consider the weighted undirected hypergraph H = (V, E , W) as a c-uniform hypergraph with a signal s ∈ R |V | where c is an even number. If c is odd, we preprocess all hyperedges by adding an auxiliary vertex to each of them to make their cardinalities even in our previous work [36]. Specifically, we denote the preprocessed hypergraph by H = (V , E , W) where the weight matrix remains unchanged.…”