A General Latent Embedding Approach for Modeling Non-uniform High-dimensional Sparse Hypergraphs with Multiplicity

2024-10-15

AI Summary

Existing hypergraph modeling approaches critically rely on uniform hyperedge orders, neglect multiedges, and fail to capture high-dimensional sparse heterogeneity. Method: This paper proposes the first unified latent-variable embedding framework that jointly handles variable hyperedge orders and multiplicities. Theoretically, we rigorously establish parameter identifiability, estimation convergence rate, and asymptotic normality. Methodologically, we integrate vertex-degree heterogeneity, order-adaptive regularization, and latent embeddings, and design a projection gradient ascent algorithm for efficient optimization. Contribution/Results: Extensive simulations confirm theoretical consistency. On real-world co-citation hypergraphs, our method significantly improves structural pattern discovery accuracy and link prediction performance, overcoming fundamental expressivity limitations of conventional hypergraph models.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Modeling non-uniform high-dimensional sparse hypergraphs with multiplicity
• Overcoming limitations of uniformity restrictions in hyperlink orders
• Accounting for repeated observations of identical hyperlinks

Innovation

• Latent embeddings model non-uniform hypergraphs with multiplicity
• Vertex degree heterogeneity and order-adjusting parameters address limitations
• Projected gradient ascent algorithm enables effective parameter estimation