A unified framework of principal component analysis and factor analysis

Abstract

Principal component analysis and factor analysis are fundamental multivariate analysis methods. In this paper a unified framework to connect them is introduced. Under a general latent variable model, we present matrix optimization problems from the viewpoint of loss function minimization, and show that the two methods can be viewed as solutions to the optimization problems with specific loss functions. Specifically, principal component analysis can be derived from a broad class of loss functions including the ${\ell }_2$ norm, while factor analysis corresponds to a modified ${\ell }_0$ norm problem. Related problems are discussed, including algorithms, penalized maximum likelihood estimation under the latent variable model, and a principal component factor model. These results can lead to new tools of data analysis and research topics.