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We study streaming principal component analysis (PCA), that is to find, in$O(dk)$ space, the top $k$ eigenvectors of a $d\times d$ hidden matrix $\bf\Sigma$ with online vectors drawn from covariance matrix $\bf \Sigma$.  We provide $\textit{global}$ convergence for Oja's algorithm which ispopularly used in practice but lacks theoretical understanding for $k>1$. Wealso provide a modified variant $\mathsf{Oja}^{++}$ that runs $\textit{evenfaster}$ than Oja's. Our results match the information theoretic lower bound interms of dependency on error, on eigengap, on rank $k$, and on dimension $d$,up to poly-log factors. In addition, our convergence rate can be made gap-free,that is proportional to the approximation error and independent of theeigengap.  In contrast, for general rank $k$, before our work (1) it was open to designany algorithm with efficient global convergence rate; and (2) it was open todesign any algorithm with (even local) gap-free convergence rate in $O(dk)$space.

First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and Near-Optimal Rate