A unified framework of principal component analysis and factor analysis

TL;DR

This paper introduces a unified framework linking principal component analysis and factor analysis through matrix optimization problems based on different loss functions, enhancing understanding and potential applications.

Contribution

It presents a novel unified framework connecting PCA and factor analysis via a general latent variable model and specific loss functions, offering new analytical tools.

Findings

PCA derived from broad loss functions including L2 norm

Factor analysis corresponds to a modified L0 norm problem

Framework enables new data analysis tools and research directions

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 L2 norm, while factor analysis corresponds to a modified L0 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.