[2510.14523] On the Identifiability of Tensor Ranks via Prior Predictive Matching

Computer Science > Machine Learning arXiv:2510.14523 (cs) [Submitted on 16 Oct 2025 (v1), last revised 1 Apr 2026 (this version, v2)] Title:On the Identifiability of Tensor Ranks via Prior Predictive Matching Authors:Eliezer da Silva, Arto Klami, Diego Mesquita, Iñigo Urteaga View a PDF of the paper titled On the Identifiability of Tensor Ranks via Prior Predictive Matching, by Eliezer da Silva and 3 other authors View PDF HTML (experimental) 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 ba...