Recent research has shown growing interest in modeling hypergraphs, which capture polyadic interactions among entities beyond traditional dyadic relations. However, most existing methodologies for hypergraphs face significant limitations, including their heavy reliance on uniformity restrictions for hyperlink orders and their inability to account for repeated observations of identical hyperlinks. In this work, we introduce a novel and general latent embedding approach that addresses these challenges through the integration of latent embeddings, vertex degree heterogeneity parameters, and an order-adjusting parameter. Theoretically, we investigate the identifiability conditions for the latent embeddings and associated parameters, and we establish the convergence rates of their estimators along with asymptotic distributions. Computationally, we employ a projected gradient ascent algorithm for parameter estimation. Comprehensive simulation studies demonstrate the effectiveness of the algorithm and validate the theoretical findings. Moreover, an application to a co-citation hypergraph illustrates the advantages of the proposed method.

翻译：近期研究对超图建模表现出日益浓厚的兴趣，超图能够捕捉超越传统二元关系之外实体间的多元交互。然而，现有的大多数超图建模方法面临显著局限，包括其严重依赖于超链接阶数的均匀性限制，以及无法处理对相同超链接的重复观测。在本工作中，我们提出了一种新颖且通用的潜在嵌入方法，该方法通过整合潜在嵌入、顶点度异质性参数以及阶数调整参数来应对这些挑战。在理论上，我们研究了潜在嵌入及相关参数的可识别性条件，并建立了其估计量的收敛速度及渐近分布。在计算上，我们采用投影梯度上升算法进行参数估计。全面的模拟研究证明了该算法的有效性并验证了理论结果。此外，在一个共引超图上的应用展示了所提方法的优势。