  Principal component analysis (PCA) is a simple and popular tool for
processing high-dimensional data. We investigate its effectiveness for matrix
denoising.
  We consider the clean data are generated from a low-dimensional subspace, but
masked by independent high-dimensional sub-Gaussian noises with standard
deviation $\sigma$. Under the low-rank assumption on the clean data with a mild
spectral gap assumption, we prove that the distance between each pair of
PCA-denoised data point and the clean data point is uniformly bounded by
$O(\sigma \log n)$. To illustrate the spectral gap assumption, we show it can
be satisfied when the clean data are independently generated with a
non-degenerate covariance matrix. We then provide a general lower bound for the
error of the denoised data matrix, which indicates PCA denoising gives a
uniform error bound that is rate-optimal. Furthermore, we examine how the error
bound impacts downstream applications such as clustering and manifold learning.
Numerical results validate our theoretical findings and reveal the importance
of the uniform error.
