How close are the eigenvectors and eigenvalues of the sample and actual covariance matrices?

Abstract: How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite second moment and distributions supported in a centered Euclidean ball, we prove that the inner product between eigenvectors of the sample and actual covariance matrices decreases proportionally to the respective eigenvalue distance. Our findings imply non-asymptotic concentration bounds for eigenvectors, eigenspaces, and eigenvalues. They also provide conditions for distinguishing principal components based on a constant number of samples.