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We analyse the reconstruction error of principal component analysis (PCA) and prove non-asymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give oracle inequalities under relative eigenvalue conditions. The bounds reveal that the excess risk differs considerably from usually considered subspace distances based on canonical angles. Our approach relies on the analysis of empirical spectral projectors combined with concentration inequalities for weighted empirical covariance operators and empirical eigenvalues.

Nonasymptotic upper bounds for the reconstruction error of PCA