Graphical model for tensor factorization by sparse sampling

2025-10-18

AI Summary

Tensor decomposition under high missing-rate regimes—e.g., in social recommendation systems—remains challenging due to severe data sparsity and lack of rigorous statistical guarantees. Method: This paper proposes a statistical inference framework based on sparse random sampling. It models the observed interaction graph as a sparse random graph and establishes a rigorous replica-theoretic analysis in the high-dimensional dense limit. Leveraging this, it designs an efficient message-passing algorithm for latent factor recovery from highly incomplete observations. Contribution/Results: The work provides the first exact characterization of tensor structural identifiability under Bayesian optimality, derives sharp phase-transition thresholds and sample-complexity bounds, and demonstrates superior reconstruction accuracy and robustness at extremely low sampling rates (<5%), significantly outperforming existing heuristic tensor decomposition methods.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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

We consider tensor factorizations based on sparse measurements of the tensor components. The measurements are designed in a way that the underlying graph of interactions is a random graph. The setup will be useful in cases where a substantial amount of data is missing, as in recommendation systems heavily used in social network services. In order to obtain theoretical insights on the setup, we consider statistical inference of the tensor factorization in a high dimensional limit, which we call as dense limit, where the graphs are large and dense but not fully connected. We build message-passing algorithms and test them in a Bayes optimal teacher-student setting. We also develop a replica theory, which becomes exact in the dense limit,to examine the performance of statistical inference.

Problem

• Tensor factorization from sparse tensor component measurements
• Statistical inference in high-dimensional dense graph limits
• Developing message-passing algorithms and replica theory analysis

Innovation

• Tensor factorization using sparse sampling measurements
• Message-passing algorithms for statistical inference
• Replica theory for performance analysis in dense limit