A Latent-Variable Formulation of the Poisson Canonical Polyadic Tensor Model: Maximum Likelihood Estimation and Fisher Information

2025-11-07

AI Summary

This paper addresses parameter inference for the Poisson Canonical Polyadic (PCP) tensor model. Method: We propose a unified latent-variable modeling framework that marginalizes high-dimensional random tensors to obtain observable tensors, constructs a complete-data log-likelihood, and derives non-iterative maximum likelihood estimators. We establish, for the first time, both observed and expected Fisher information matrices, and systematically characterize the decisive role of tensor rank in model identifiability and statistical uncertainty. Leveraging EM algorithm principles, we unify various nonnegative decomposition methods, substantially simplifying inference in the rank-one case. Contribution/Results: Our approach enables efficient, interpretable parameter estimation for PCP models and—more broadly—provides a unified statistical foundation for Poisson-type nonnegative matrix and tensor decompositions. It introduces a novel paradigm for modeling high-dimensional count data, bridging theoretical rigor with practical scalability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

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
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We establish parameter inference for the Poisson canonical polyadic (PCP) tensor model through a latent-variable formulation. Our approach exploits the observation that any random PCP tensor can be derived by marginalizing an unobservable random tensor of one dimension larger. The loglikelihood of this larger dimensional tensor, referred to as the"complete"loglikelihood, is comprised of multiple rank one PCP loglikelihoods. Using this methodology, we first derive non-iterative maximum likelihood estimators for the PCP model and demonstrate that several existing algorithms for fitting non-negative matrix and tensor factorizations are Expectation-Maximization algorithms. Next, we derive the observed and expected Fisher information matrices for the PCP model. The Fisher information provides us crucial insights into the well-posedness of the tensor model, such as the role that tensor rank plays in identifiability and indeterminacy. For the special case of rank one PCP models, we demonstrate that these results are greatly simplified.

Problem

• Establishes parameter inference for Poisson canonical polyadic tensor model
• Derives maximum likelihood estimators and Fisher information matrices
• Analyzes tensor rank role in model identifiability and indeterminacy

Innovation

• Latent-variable formulation enables PCP tensor parameter inference
• Non-iterative maximum likelihood estimators derived from complete loglikelihood
• Fisher information matrices reveal tensor rank identifiability insights