The construction of a meaningful hypergraph topology is the key to processing signals with high-order relationships that involve more than two entities. Learning the hypergraph structure from the observed signals to capture the intrinsic 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, to address 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 learning framework 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 the proposed framework on both synthetic and real world datasets. Experiments show that our proposed framework can efficiently infer meaningful hypergraph topologies from observed signals.

翻译：构建有意义的超图拓扑结构是利用高阶关系（涉及两个以上实体）处理信号的关键。当数据集中未提供现成超图拓扑时，从观测信号中学习超图结构以捕获实体间的内在关系变得至关重要。该问题的核心存在两个挑战：1) 如何处理潜在超边的巨大搜索空间，2) 如何定义有意义的准则来衡量节点上观测信号与超图结构之间的关系。本文为应对第一个挑战，假设理想超图结构可从捕获信号内成对关系的可学习图结构推导而来。进一步，我们提出了一种具有新颖双重平滑先验的超图学习框架，该先验揭示了观测节点信号与超图结构之间的映射关系，其中每个超边对应可学习图结构中兼具节点信号平滑性和边信号平滑性的子图。最后，我们在合成数据集和真实数据集上进行了广泛实验来评估所提框架。实验表明，我们的框架能够从观测信号中高效推断出有意义的超图拓扑结构。