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How many samples are sufficient to guarantee that the eigenvectors andeigenvalues of the sample covariance matrix are close to those of the actualcovariance matrix? For a wide family of distributions, including distributionswith finite second moment and distributions supported in a centered Euclideanball, we prove that the inner product between eigenvectors of the sample andactual covariance matrices decreases proportionally to the respectiveeigenvalue distance. Our findings imply non-asymptotic concentration bounds foreigenvectors, eigenspaces, and eigenvalues. They also provide conditions fordistinguishing principal components based on a constant number of samples.

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