Scalable Hypergraph Structure Learning with Diverse Smoothness Priors

In graph signal processing, learning the weighted connections between nodes from a set of sample signals is a fundamental task when the underlying relationships are not known a priori. This task is typically addressed by finding a graph Laplacian on which the observed signals are smooth. With the extension of graphs to hypergraphs - where edges can connect more than two nodes - graph learning methods have similarly been generalized to hypergraphs. However, the absence of a unified framework for calculating total variation has led to divergent definitions of smoothness and, consequently, differing approaches to hyperedge recovery. We confront this challenge through generalization of several previously proposed hypergraph total variations, subsequently allowing ease of substitution into a vector based optimization. To this end, we propose a novel hypergraph learning method that recovers a hypergraph topology from time-series signals based on a smoothness prior. Our approach, designated as Hypergraph Structure Learning with Smoothness (HSLS), addresses key limitations in prior works, such as hyperedge selection and convergence issues, by formulating the problem as a convex optimization solved via a forward-backward-forward algorithm, ensuring guaranteed convergence. Additionally, we introduce a process that simultaneously limits the span of the hyperedge search and maintains a valid hyperedge selection set. In doing so, our method becomes scalable in increasingly complex network structures. The experimental results demonstrate improved performance, in terms of accuracy, over other state-of-the-art hypergraph inference methods; furthermore, we empirically show our method to be robust to total variation terms, biased towards global smoothness, and scalable to larger hypergraphs.