Dimension estimation in noisy PCA with SURE and random matrix theory

Principal component analysis (PCA) is one of the best known methods for dimensionality reduction. Perhaps the most important problem in using PCA is to determine the number of principal components (PCs) or equivalently choose the rank of the loading matrix. Many methods have been proposed to deal with this problem but almost all of them fail in the important practical case when the number of observation is comparable to the number of variables, i.e., the realm of random matrix theory (RMT). In this paper we propose to use Stein's unbiased risk estimator (SURE) to estimate, with some assistance from RMT, the number of principal components. The method is applied both on simulated and real functional magnetic resonance imaging (fMRI) data, and compared to BIC and the Laplace method.