Recently, hypergraphs have attracted a lot of attention due to their ability to capture complex relations among entities. The insurgence of hypergraphs has resulted in data of increasing size and complexity that exhibit interesting small-scale and local structure, e.g., small-scale communities and localized node-ranking around a given set of seed nodes. Popular and principled ways to capture the local structure are the local hypergraph clustering problem and related seed set expansion problem. In this work, we propose the first local diffusion method that achieves edge-size-independent Cheeger-type guarantee for the problem of local hypergraph clustering while applying to a rich class of higher-order relations that covers many previously studied special cases. Our method is based on a primal-dual optimization formulation where the primal problem has a natural network flow interpretation, and the dual problem has a cut-based interpretation using the $\ell_2$-norm penalty on associated cut-costs. We demonstrate the new technique is significantly better than state-of-the-art methods on both synthetic and real-world data.
翻译:最近,高音由于能够捕捉各实体之间的复杂关系而引起人们的极大关注。高音暴发后,出现了规模和复杂性不断增大的数据,显示出有趣的小型和地方结构,例如小型社区和围绕一组种子节点的局部节点。捕捉当地结构的流行和有原则的方法是当地高音集聚问题和相关的种子集成扩展问题。在这项工作中,我们提议了第一个本地传播方法,即对本地高音集成问题实现边缘大小独立的Cheeger型担保,同时对以前研究过的许多特殊案例适用丰富的高音集成型关系。我们的方法基于原始-双优化配制,其中原始问题具有自然网络流动解释,而双重问题则使用对相关削减成本的$_2美元-诺姆罚款进行剪切解释。我们证明新技术比合成和现实世界数据方面的最新方法要好得多。