Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and
  Near-Optimal Rate   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 is
popularly used in practice but lacks theoretical understanding for $k>1$. We
also provide a modified variant $\mathsf{Oja}^{++}$ that runs $\textit{even
faster}$ than Oja's. Our results match the information theoretic lower bound in
terms 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 the
eigengap.
  In contrast, for general rank $k$, before our work (1) it was open to design
any algorithm with efficient global convergence rate; and (2) it was open to
design any algorithm with (even local) gap-free convergence rate in $O(dk)$
space.

27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph