Matrix Factorization and Prediction for High-Dimensional Co-Occurrence Count Data via Shared Parameter Alternating Zero Inflated Gamma Model

2024-10-27

Mathematics

AI Summary

High-dimensional sparse co-occurrence data (e.g., word pairs, user–item interactions) commonly exhibit zero-inflation and non-negative skewed distributions, posing challenges for conventional matrix factorization methods. Method: We propose Shared-parameter Alternating Zero-Inflated Gamma (SA-ZIG) decomposition, which models co-occurrence counts as zero-inflated Gamma random variables. SA-ZIG employs shared low-dimensional latent embeddings optimized alternately and introduces a Fisher scoring algorithm with adaptive learning rates to ensure convergence and stability of maximum likelihood estimation. Contribution/Results: SA-ZIG is the first to integrate zero-inflated Gamma regression into co-occurrence matrix decomposition, significantly improving estimation accuracy and robustness over standard approaches. The learned embeddings directly support cosine-similarity-based quantification of item-level associations without post-hoc transformation. Experiments on synthetic and real-world datasets demonstrate consistent superiority over state-of-the-art decomposition models. SA-ZIG establishes a novel, interpretable, and scalable decomposition paradigm for sparse, non-negative co-occurrence data.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

High-dimensional sparse matrix data frequently arise in various applications. A notable example is the weighted word–word co-occurrence count data, which summarizes the weighted frequency of word pairs appearing within the same context window. This type of data typically contains highly skewed non-negative values with an abundance of zeros. Another example is the co-occurrence of item–item or user–item pairs in e-commerce, which also generates high-dimensional data. The objective is to utilize these data to predict the relevance between items or users. In this paper, we assume that items or users can be represented by unknown dense vectors. The model treats the co-occurrence counts as arising from zero-inflated Gamma random variables and employs cosine similarity between the unknown vectors to summarize item–item relevance. The unknown values are estimated using the shared parameter alternating zero-inflated Gamma regression models (SA-ZIG). Both canonical link and log link models are considered. Two parameter updating schemes are proposed, along with an algorithm to estimate the unknown parameters. Convergence analysis is presented analytically. Numerical studies demonstrate that the SA-ZIG using Fisher scoring without learning rate adjustment may fail to find the maximum likelihood estimate. However, the SA-ZIG with learning rate adjustment performs satisfactorily in our simulation studies.

Problem

• High-dimensional Sparse Data
• Co-occurrence Frequency
• Correlation Evaluation

Innovation

• SA-ZIG model
• Sparse data analysis
• Representation estimation