  Principal component analysis (PCA) has been widely used in analyzing
high-dimensional data. It converts a set of observed data points of possibly
correlated variables into a set of linearly uncorrelated variables via an
orthogonal transformation. To handle streaming data and reduce the complexities
of PCA, (subspace) online PCA iterations were proposed to iteratively update
the orthogonal transformation by taking one observed data point at a time.
Existing works on the convergence of (subspace) online PCA iterations mostly
focus on the case where sample are almost surely uniformly bounded. In this
paper, we analyze the convergence of a subspace online PCA iteration under more
practical assumption and obtain a nearly optimal finite-sample error bound. Our
convergence rate almost matches the minimax information lower bound. We prove
that the convergence is nearly global in the sense that the subspace online PCA
iteration is convergent with high probability for random initial guesses. This
work also leads to a simpler proof of the recent work on analyzing online PCA
for the first principal component only.
