We propose a new matrix factor model, named RaDFaM, the latent structure of which is strictly derived based on a hierarchical rank decomposition of a matrix. Hierarchy is in the sense that all basis vectors of the column space of each multiplier matrix are assumed the structure of a vector factor model. Compared to the most commonly used matrix factor model that takes the latent structure of a bilinear form, RaDFaM involves additional row-wise and column-wise matrix latent factors. This yields modest dimension reduction and stronger signal intensity from the sight of tensor subspace learning, though poses challenges of new estimation procedure and concomitant inferential theory for a collection of matrix-valued observations. We develop a class of estimation procedure that makes use of the separable covariance structure under RaDFaM and approximate least squares, and derive its superiority in the merit of the peak signal-to-noise ratio. We also establish the asymptotic theory when the matrix-valued observations are uncorrelated or weakly correlated. Numerically, in terms of image/matrix reconstruction, supervised learning, and so forth, we demonstrate the excellent performance of RaDFaM through two matrix-valued sequence datasets of independent 2D images and multinational macroeconomic indices time series, respectively.

翻译：我们提出一种新的矩阵因子模型RaDFaM，其潜在结构严格基于矩阵的层次化秩分解推导。层次性体现在：每个乘子矩阵列空间的所有基向量均假设具有向量因子模型的结构。与最常采用的具有双线性形式潜在结构的矩阵因子模型相比，RaDFaM引入了额外的行向与列向矩阵潜在因子。这从张量子空间学习的视角实现了适度的降维和更强的信号强度，尽管给矩阵值观测集合的估计程序及其伴随的推断理论带来了新挑战。我们发展了一类利用RaDFaM可分离协方差结构与近似最小二乘的估计程序，并从峰值信噪比指标上证明了其优越性。同时，当矩阵值观测不相关或弱相关时，我们建立了渐近理论。在数值实验中，通过两个矩阵值序列数据集（独立二维图像与跨国宏观经济指标时间序列），我们从图像/矩阵重建、监督学习等方面展示了RaDFaM的优异性能。