On the Identifiability of Tensor Ranks via Prior Predictive Matching

2025-10-16

AI Summary

This paper addresses the lack of rigorous theoretical foundations for tensor rank identifiability in probabilistic tensor models. We propose a prior-predictive moment-matching framework for rank selection. Our key contribution is the first formal equivalence established between rank identifiability and the solvability of a log-linear moment equation system, yielding a closed-form rank estimator that requires no model training. By analyzing marginal moments, we transform moment-matching conditions into analytic equations involving hyperparameters, observed moments, and rank; leveraging algebraic graph theory, we prove that PARAFAC/CP, Tensor Train, and Tensor Ring models satisfy identifiability conditions due to their topological structures, whereas the Tucker model is inherently unidentifiable. Experiments on synthetic and real-world datasets demonstrate that our estimator is accurate, robust, and computationally efficient. This work provides the first theoretically sound, model-agnostic paradigm for rank inference in probabilistic tensor modeling.

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: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Selecting the latent dimensions (ranks) in tensor factorization is a central challenge that often relies on heuristic methods. This paper introduces a rigorous approach to determine rank identifiability in probabilistic tensor models, based on prior predictive moment matching. We transform a set of moment matching conditions into a log-linear system of equations in terms of marginal moments, prior hyperparameters, and ranks; establishing an equivalence between rank identifiability and the solvability of such system. We apply this framework to four foundational tensor-models, demonstrating that the linear structure of the PARAFAC/CP model, the chain structure of the Tensor Train model, and the closed-loop structure of the Tensor Ring model yield solvable systems, making their ranks identifiable. In contrast, we prove that the symmetric topology of the Tucker model leads to an underdetermined system, rendering the ranks unidentifiable by this method. For the identifiable models, we derive explicit closed-form rank estimators based on the moments of observed data only. We empirically validate these estimators and evaluate the robustness of the proposal.

Problem

• Determining identifiable tensor ranks via moment matching conditions
• Establishing equivalence between rank identifiability and equation solvability
• Deriving closed-form rank estimators for identifiable tensor models

Innovation

• Determines tensor rank identifiability via moment matching
• Transforms conditions into solvable log-linear equation systems
• Derives closed-form rank estimators from data moments