We propose MNPCA, a novel non-linear generalization of (2D)$^2${PCA}, a classical linear method for the simultaneous dimension reduction of both rows and columns of a set of matrix-valued data. MNPCA is based on optimizing over separate non-linear mappings on the left and right singular spaces of the observations, essentially amounting to the decoupling of the two sides of the matrices. We develop a comprehensive theoretical framework for MNPCA by viewing it as an eigenproblem in reproducing kernel Hilbert spaces. We study the resulting estimators on both population and sample levels, deriving their convergence rates and formulating a coordinate representation to allow the method to be used in practice. Simulations and a real data example demonstrate MNPCA's good performance over its competitors.

翻译：我们提出MNPCA，这是一种针对(2D)²PCA的非线性推广方法，(2D)²PCA是一种经典线性方法，用于同时对矩阵数据集的各行和各列进行降维。MNPCA基于对观测数据左右奇异空间分别优化非线性映射，本质上实现了矩阵两侧的解耦。我们通过将MNPCA视为再生核希尔伯特空间中的特征问题，为其建立了全面的理论框架。我们在总体和样本层面研究所得估计量，推导其收敛速率，并构建坐标表示以便该方法在实际中应用。模拟实验和真实数据案例表明，MNPCA相比其他方法具有更优性能。