标题： 通过先验预测匹配的张量秩可识别性 显示英文标题

标题： On the Identifiability of Tensor Ranks via Prior Predictive Matching

摘要： 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。 本文介绍了一种严格的方法，用于确定概率张量模型中的秩可识别性，该方法基于先验预测矩匹配。 我们将一组矩匹配条件转换为关于边缘矩、先验超参数和秩的对数线性方程组；建立了秩可识别性与该系统可解性之间的等价关系。 我们将这一框架应用于四个基础张量模型，证明PARAFAC/CP模型的线性结构、Tensor Train模型的链式结构以及Tensor Ring模型的闭环结构会产生可解系统，使其秩具有可识别性。 相反，我们证明Tucker模型的对称拓扑会导致欠定系统，使得通过此方法无法识别其秩。 对于可识别模型，我们仅基于观测数据的矩推导出显式的闭合形式秩估计器。 我们对这些估计器进行了经验验证，并评估了该方法的鲁棒性。 显示英文摘要

摘要： 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.