  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 mild eigenvalue conditions. The
bounds reveal that the excess risk differs significantly 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.
