通过先验预测匹配，解决张量分解中秩的可识别性问题。
On the Identifiability of Tensor Ranks via Prior Predictive Matching
- 基于先验预测矩匹配，将秩可识别性转化为可解方程组。
- 证明CP、TT、TR模型秩可识别，而Tucker模型不可识别。
- 给出仅依赖观测数据矩的秩估计闭式解，适合贝叶斯张量建模者。
选择张量分解中的潜在维度（秩）是一个核心挑战，通常依赖启发式方法。本文提出一种严格方法，基于先验预测矩匹配来确定概率张量模型中秩的可识别性。将一组矩匹配条件转化为关于边际矩、先验超参数和秩的对数线性方程组，建立秩可识别性与该系统可解性的等价关系。将该框架应用于四种基础张量模型：发现PARAFAC/CP模型的线性结构、张量列车（Tensor Train）的链式结构以及张量环（Tensor Ring）的闭环结构均能导出可解系统，使其秩可识别；而Tucker模型的对称拓扑导致方程组欠定，秩不可识别。对于可识别模型，我们推导出仅依赖观测数据矩的显式闭式秩估计器，并通过实验验证其有效性与鲁棒性。
原文摘要 · Abstract (English)
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.
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