On the Identifiability of Tensor Ranks via Prior Predictive Matching
This provides a theoretical foundation for rank selection in tensor models, addressing a central challenge in machine learning and data analysis, though it is incremental as it builds on existing tensor factorization frameworks.
The paper tackled the problem of selecting latent dimensions (ranks) in tensor factorization by introducing a rigorous approach based on prior predictive moment matching, proving that ranks are identifiable for PARAFAC/CP, Tensor Train, and Tensor Ring models but not for the Tucker model, and deriving explicit closed-form rank estimators validated empirically.
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.