Faster Low-rank Approximation using Adaptive Gap-based Preconditioning.
arXiv: Information Theory(2016)
摘要
We propose a method for rank $k$ approximation to a given input matrix $X mathbb{R}^{d times n}$ which runs in time [ tilde{O} left(d ~cdot~ minleft{n + tilde{sr}(X) ,G^{-2}_{k,p+1} , n^{3/4}, tilde{sr}(X)^{1/4} ,G^{-1/2}_{k,p+1} right} ~cdot~ text{poly}(p)right) ~, ] where $pu003ek$, $tilde{sr}(X)$ is related to stable rank of $X$, and $G_{k,p+1} = frac{sigma_k-sigma_p}{sigma_k}$ is the multiplicative gap between the $k$-th and the $(p+1)$-th singular values of $X$. In particular, this yields a linear time algorithm if the gap is at least $1/sqrt{n}$ and $k,p,tilde{sr}(X)$ are constants.
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