Information inequalities for the estimation of principal components
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We provide lower bounds for the estimation of the eigenspaces of a covariance operator. These information inequalities are non-asymptotic and can be applied to any sequence of eigenvalues. In the important case of the eigenspace of the d leading eigenvalues, the lower bounds match non-asymptotic upper bounds based on the empirical covariance operator. Our approach relies on a van Trees inequality for equivariant models, with the reference measure being the Haar measure on the orthogonal group, combined with large deviations techniques to design optimal prior densities.
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