  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.
