A generalized information criterion for high-dimensional PCA rank selection

Principal component analysis (PCA) is the most commonly used statistical procedure for dimension reduction. An important issue for applying PCA is to determine the rank, which is the number of dominant eigenvalues of the covariance matrix. The Akaike information criterion (AIC) and Bayesian information criterion (BIC) are among the most widely used rank selection methods. Both use the number of free parameters for assessing model complexity. In this work, we adopt the generalized information criterion (GIC) to propose a new method for PCA rank selection under the high-dimensional framework. The GIC model complexity takes into account the sizes of covariance eigenvalues and can be better adaptive to practical applications. Asymptotic properties of GIC are derived and the selection consistency is established under the generalized spiked covariance model.