Even Faster SVD Decomposition Yet Without Agonizing Pain

We study k-SVD that is to obtain the first k singular vectors of a matrix A approximately. Recently, a few breakthroughs have been discovered on k-SVD: Musco and Musco [18] provided the first gap-free theorem for the block Krylov method, Shamir [20] discovered the first variance-reduction stochastic method, and Bhojanapalli et al. [6] provided the fastest O(nnz(A) + poly(1/ε))-type of algorithm using alternating minimization. In this paper, put forward a new framework for SVD and improve the above breakthroughs. We obtain faster gap-free convergence rate outperforming [18], we obtain the first accelerated and stochastic method outperforming [20]. In the O(nnz(A) + poly(1/ε)) running-time regime, we outperform [6] in certain parameter regimes without even using alternating minimization.