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增长的兴趣，超图能够捕捉超越传统二元关系的实体间多元交互。然而，现有的大多数超图建模方法面临显著局限性，包括对超链接阶数的均匀性限制的严重依赖，以及无法处理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，通过整合潜在嵌入、顶点度异质性参数和阶数调整参数来解决这些挑战。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率及渐近分布。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，在共引超图上的应用展示了所提方法的优势。