On the Identifiability of Tensor Ranks via Prior Predictive Matching

Eliezer de Souza da Silva, Arto Klami, Diego Mesquita, Iñigo Urteaga

TL;DR: The identification of ranks in probabilistic tensor factorization using a prior predictive moments-matching method is identifiable for PARAFAC/CP and Tensor Train models, but not for the Tucker model.

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.

Submission Number: 2287