High dimensional data and systems with many degrees of freedom are often characterized by covariance matrices. In this paper, we consider the problem of simultaneously estimating the dimension of the principal (dominant) subspace of these covariance matrices and obtaining an approximation to the subspace. This problem arises in the popular principal component analysis (PCA), and in many applications of machine learning, data analysis, signal and image processing, and others. We first present a novel method for estimating the dimension of the principal subspace. We then show how this method can be coupled with a Krylov subspace method to simultaneously estimate the dimension and obtain an approximation to the subspace. The dimension estimation is achieved at no additional cost. The proposed method operates on a model selection framework, where the novel selection criterion is derived based on random matrix perturbation theory ideas. We present theoretical analyses which (a) show that th 查看全文>>