Linear Dimensionality Reduction: Survey, Insights, and Generalizations

TLDR

Linear dimensionality reduction methods are essential for analyzing high‑dimensional data, capturing features such as covariance, dynamics, correlations, input‑output relations, and class margins, yet their diverse names and motivations have obscured connections across fields. The authors aim to survey these diverse methods as optimization programs over matrix manifolds and to demonstrate how this framework enables generalizations and novel variants. They present a generic optimization framework over matrix manifolds that unifies PCA, factor analysis, MDS, LDA, CCA, ICA, regression, metric learning, and others, and implement a solver that produces optimal low‑dimensional projections and allows construction of new variants such as orthogonal‑projection CCA. The survey reveals shortcomings of traditional eigenvector solutions, shows that the framework can act as a black‑box, objective‑agnostic tool, and demonstrates new variants, indicating that linear dimensionality reduction can evolve into a versatile numerical technology.

Abstract

Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of interest, such as covariance, dynamical structure, correlation between data sets, input-output relationships, and margin between data classes. Methods have been developed with a variety of names and motivations in many fields, and perhaps as a result the connections between all these methods have not been highlighted. Here we survey methods from this disparate literature as optimization programs over matrix manifolds. We discuss principal component analysis, factor analysis, linear multidimensional scaling, Fisher's linear discriminant analysis, canonical correlations analysis, maximum autocorrelation factors, slow feature analysis, sufficient dimensionality reduction, undercomplete independent component analysis, linear regression, distance metric learning, and more. This optimization framework gives insight to some rarely discussed shortcomings of well-known methods, such as the suboptimality of certain eigenvector solutions. Modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver, which accepts as input data and an objective to be optimized, and returns, as output, an optimal low-dimensional projection of the data. This simple optimization framework further allows straightforward generalizations and novel variants of classical methods, which we demonstrate here by creating an orthogonal-projection canonical correlations analysis. More broadly, this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox, objective-agnostic numerical technology.