A Unified Trace-Optimization Framework for Multidimensionality Reduction

TL;DR

This paper unifies various multidimensionality reduction techniques within a trace optimization framework, comparing their theoretical foundations, assumptions, and practical applications, including linear and kernel-based nonlinear methods.

Contribution

It introduces a comprehensive unified framework for multidimensionality reduction methods, including linear and nonlinear extensions, with comparative analysis and practical guidelines.

Findings

Unified trace optimization framework for multiple methods

Kernel extensions enable capturing nonlinear relations

Guidelines for selecting appropriate techniques

Abstract

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.