Structure-Preserving Non-Linear PCA for Matrices

We propose a new dimension reduction method for matrix-valued data called Matrix Non-linear PCA (MNPCA), which is a non-linear generalization of (2D)${}^{2}$PCA. 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.