A Unified Trace-Optimization Framework for Multidimensionality Reduction

This paper presents a comprehensive overview of several multidimensional reduction methods focusing on Multidimensional Principal Component Analysis (MPCA), Multilinear Orthogonal Neighborhood Preserving Projection (MONPP), Multidimensional Locally Linear Embedding (MLLE), and Multidimensional Laplacian Eigenmaps (MLE). These techniques are formulated within a unified framework based on trace optimization, where the dimensionality reduction problem is expressed as maximization or minimization problems. In addition to the linear MPCA and MONPP approaches, kernel-based extensions of these methods also are presented. The latter methods make it possible to capture nonlinear relations between high-dimensional data. A comparative analysis highlights the theoretical foundations, assumptions, and computational efficiency of each method, as well as their practical applicability. The study provides insights and guidelines for selecting an appropriate dimensionality reduction technique suited to the application at hand.