Structure-preserving non-linear PCA for matrices

TL;DR

MNPCA is a new non-linear extension of 2D PCA that reduces dimensions of matrix data by optimizing separate non-linear mappings for rows and columns, with theoretical guarantees and practical effectiveness.

Contribution

It introduces MNPCA, a novel non-linear method for matrix data dimension reduction, with a comprehensive theoretical framework and practical implementation details.

Findings

MNPCA outperforms competitors in simulations.

Theoretical convergence rates are established.

Real data example confirms effectiveness.

Abstract

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.