Learning Hypergraphs From Signals With Dual Smoothness Prior

TL;DR

The paper addresses learning hypergraph structures from node signals when the topology is not predefined. It introduces HGSL, a two-step framework that first learns a pairwise graph $\mathbf{W}$ from the signals using a node-signal smoothness objective, and then detects hyperedges as line-graph communities under a dual smoothness prior, avoiding the $2^{N}$ combinatorial search. The method leverages a primal-dual splitting approach for graph learning and the Leiden algorithm for line-graph community detection, achieving $O(N^2)$ and $O(M^2)$-level scalability. Empirical results on synthetic and real-world datasets show HGSL outperforms baselines in Recall, Precision, and F1, demonstrating its ability to capture meaningful high-order relationships in signals.

Abstract

Hypergraph structure learning, which aims to learn the hypergraph structures from the observed signals to capture the intrinsic high-order relationships among the entities, becomes crucial when a hypergraph topology is not readily available in the datasets. There are two challenges that lie at the heart of this problem: 1) how to handle the huge search space of potential hyperedges, and 2) how to define meaningful criteria to measure the relationship between the signals observed on nodes and the hypergraph structure. In this paper, for the first challenge, we adopt the assumption that the ideal hypergraph structure can be derived from a learnable graph structure that captures the pairwise relations within signals. Further, we propose a hypergraph structure learning framework HGSL with a novel dual smoothness prior that reveals a mapping between the observed node signals and the hypergraph structure, whereby each hyperedge corresponds to a subgraph with both node signal smoothness and edge signal smoothness in the learnable graph structure. Finally, we conduct extensive experiments to evaluate HGSL on both synthetic and real world datasets. Experiments show that HGSL can efficiently infer meaningful hypergraph topologies from observed signals.

Figures (3)

• Figure 1: A hypergraph that obeys the dual smoothness prior. $e_1$ and $e_2$ are two different hyperedges. Nodes of similar colour have similar node signals, and edges of similar colour have similar edge signals.
• Figure 2: An illustration of the line graph construction procedure.
• Figure 3: Performance of the proposed method and baselines with respect to the overlapping rate.