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. T o 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. YPERGRAPHS are considered as generalized graphs that capture higher order relationships [1]. While graphs encode pairwise relationships between nodes through edges, the higher order nature of hypergraphs extends node relations to allow an arbitrary number of nodes to be connected by a hyperedge. Figure 1 contains a sample hypergraph displaying these higher order connections where nodes are considered workers and hyperedges connect workers who are collaborating on a project. B. T. Brown, H. Zhang, and D. L. Lau are with the Department of Electrical and Computer Engineering, University of Kentucky, Lexington, KY 40506, USA. G. R. Arce is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA. This work was partially supported by the National Science Foundation under grants 1815992 and 1816003 and the AFOSR award FA9550-22-1-0362.