This paper introduces a novel framework called Mode-wise Principal Subspace Pursuit (MOP-UP) to extract hidden variations in both the row and column dimensions for matrix data. To enhance the understanding of the framework, we introduce a class of matrix-variate spiked covariance models that serve as inspiration for the development of the MOP-UP algorithm. The MOP-UP algorithm consists of two steps: Average Subspace Capture (ASC) and Alternating Projection (AP). These steps are specifically designed to capture the row-wise and column-wise dimension-reduced subspaces which contain the most informative features of the data. ASC utilizes a novel average projection operator as initialization and achieves exact recovery in the noiseless setting. We analyze the convergence and non-asymptotic error bounds of MOP-UP, introducing a blockwise matrix eigenvalue perturbation bound that proves the desired bound, where classic perturbation bounds fail. The effectiveness and practical merits of the proposed framework are demonstrated through experiments on both simulated and real datasets. Lastly, we discuss generalizations of our approach to higher-order data.

翻译：本文提出了一种名为“面向模式的主子空间追踪”（MOP-UP）的新框架，用于提取矩阵数据中行和列两个维度上的隐藏变异。为加深对该框架的理解，我们引入了一类矩阵变元尖峰协方差模型，该模型为MOP-UP算法的开发提供了灵感。MOP-UP算法包含两个步骤：平均子空间捕获（ASC）和交替投影（AP）。这些步骤专门设计用于捕获行方向和列方向的降维子空间，这些子空间包含数据中最具信息量的特征。ASC利用一种新颖的平均投影算子作为初始化，并在无噪声环境下实现精确恢复。我们分析了MOP-UP的收敛性和非渐近误差界，引入了一个分块矩阵特征值扰动界，证明了所需界值，而经典扰动界在此失效。通过模拟和真实数据集上的实验，验证了所提框架的有效性和实际优势。最后，我们讨论了该方法向高阶数据推广的可能性。