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.

翻译：在张量分解中选择潜在维度（秩）是一个核心挑战，通常依赖于启发式方法。本文提出了一种严谨的方法，基于先验预测矩匹配来确定概率张量模型中的秩可辨识性。我们将一组矩匹配条件转化为关于边缘矩、先验超参数和秩的对数线性方程组；建立了秩可辨识性与此类方程组可解性之间的等价关系。我们将此框架应用于四种基础张量模型，证明PARAFAC/CP模型的线性结构、Tensor Train模型的链式结构以及Tensor Ring模型的闭环结构均产生可解方程组，从而使其秩可辨识。相反，我们证明了Tucker模型的对称拓扑结构会导致欠定方程组，使得秩无法通过此方法辨识。对于可辨识模型，我们推导了仅基于观测数据矩的显式闭式秩估计器。我们通过实验验证了这些估计器，并评估了该方法的鲁棒性。