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Principal Component Analysis (PCA) is a powerful tool in statistics andmachine learning. While existing study of PCA focuses on the recovery ofprincipal components and their associated eigenvalues, there are few precisecharacterizations of individual principal component scores that yieldlow-dimensional embedding of samples. That hinders the analysis of variousspectral methods. In this paper, we first develop an $\ell_p$ perturbationtheory for a hollowed version of PCA in Hilbert spaces which provably improvesupon the vanilla PCA in the presence of heteroscedastic noises. Through a novel$\ell_p$ analysis of eigenvectors, we investigate entrywise behaviors ofprincipal component score vectors and show that they can be approximated bylinear functionals of the Gram matrix in $\ell_p$ norm, which includes $\ell_2$and $\ell_\infty$ as special examples. For sub-Gaussian mixture models, thechoice of $p$ giving optimal bounds depends on the signal-to-noise ratio, whichfurther yields optimality guarantees for spectral clustering. For contextualcommunity detection, the $\ell_p$ theory leads to a simple spectral algorithmthat achieves the information threshold for exact recovery. These also provideoptimal recovery results for Gaussian mixture and stochastic block models asspecial cases.

An $\ell_p$ theory of PCA and spectral clustering