We propose a method for rank $k$ approximation to a given input matrix $X \in \mathbb{ R}^{ d \times n}$ which runs in time \[ \tilde{ O} \left(d ~\cdot~ \min\left\{ n + \tilde{ sr}(X) \,G^{ -2}_{ k,p+1}\ ,\ n^{ 3/4}\, \tilde{ sr}(X)^{ 1/4} \,G^{ -1/2}_{ k,p+1} \right\} ~\cdot~ \text{ poly}(p)\right) ~, \] where $p>k$, $\tilde{ sr}(X)$ 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.
Categories: cs.IT math.IT