Faster Low-rank Approximation using Adaptive Gap-based Preconditioning

Alon Gonen; Shai Shalev-Shwartz

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arXiv · 1607.02925

Faster Low-rank Approximation using Adaptive Gap-based Preconditioning

Alon Gonen·Shai Shalev-Shwartz

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Abstract

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 $G_{k,p+1} = \frac{σ_k-σ_p}{σ_k}$ 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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Alon Gonen, Shai Shalev-Shwartz. 2016-07-11. Faster Low-rank Approximation using Adaptive Gap-based Preconditioning. https://arxiv.org/abs/1607.02925

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