We propose a new matrix factor model, named RaDFaM, which is strictly derived based on the general rank decomposition and assumes a structure of a high-dimensional vector factor model for each basis vector. RaDFaM contributes a novel class of low-rank latent structure that makes tradeoff between signal intensity and dimension reduction from the perspective of tensor subspace. Based on the intrinsic separable covariance structure of RaDFaM, for a collection of matrix-valued observations, we derive a new class of PCA variants for estimating loading matrices, and sequentially the latent factor matrices. The peak signal-to-noise ratio of RaDFaM is proved to be superior in the category of PCA-type estimations. We also establish the asymptotic theory including the consistency, convergence rates, and asymptotic distributions for components in the signal part. Numerically, we demonstrate the performance of RaDFaM in applications such as matrix reconstruction, supervised learning, and clustering, on uncorrelated and correlated data, respectively.

翻译：本文提出一种新的矩阵因子模型——RaDFaM，该模型基于一般秩分解严格推导，并假设每个基向量具有高维向量因子模型的结构。RaDFaM贡献了一类新颖的低秩隐式结构，从张量子空间的角度在信号强度与降维之间实现权衡。基于RaDFaM内在的可分离协方差结构，针对矩阵观测值集合，我们推导出一类新的主成分分析变体用于估计载荷矩阵，进而估计隐因子矩阵。RaDFaM的峰值信噪比被证明在PCA类估计中具有优越性。我们还建立了渐近理论，包括信号分量的一致性、收敛速率及渐近分布。在数值实验中，我们分别在非相关数据和相关数据上展示了RaDFaM在矩阵重构、监督学习及聚类等应用中的性能。