  This work obtains novel finite sample guarantees for Principal Component
Analysis (PCA). These hold even when the corrupting noise is non-isotropic, and
a part (or all of it) is data-dependent. Because of the latter, in general, the
noise and the true data are correlated. The results in this work are a
significant improvement over those given in our earlier work where this
"correlated-PCA" problem was first studied. In fact, in certain regimes, our
results imply that the sample complexity required to achieve subspace recovery
error that is a constant fraction of the noise level is near-optimal. Useful
corollaries of our result include guarantees for PCA in sparse data-dependent
noise and for PCA with missing data. An important application of the former is
in proving correctness of the subspace update step of a popular online
algorithm for dynamic robust PCA.
