Randomized SVD has become an extremely successful approach for efficiently computing a low-rank approximation of matrices. In particular the paper by Halko, Martinsson, and Tropp (SIREV 2011) contains extensive analysis, and has made it a very popular method. The typical complexity for a rank-$r$ approximation of $m\times n$ matrices is $O(mn\log n+(m+n)r^2)$ for dense matrices. The classical Nystr{\"o}m method is much faster, but applicable only to positive semidefinite matrices. This work studies a generalization of Nystr{\"o}m method applicable to general matrices, and shows that (i) it has near-optimal approximation quality comparable to competing methods, (ii) the computational cost is the near-optimal $O(mn\log n+r^3)$ for dense matrices, with small hidden constants, and (iii) crucially, it can be implemented in a numerically stable fashion despite the presence of an ill-conditioned pseudoinverse. Numerical experiments illustrate that generalized Nystr{\"o}m can significantly outperform state-of-the-art methods, especially when $r\gg 1$, achieving up to a 10-fold speedup. The method is also well suited to updating and downdating the matrix.

中文翻译：

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快速稳定的随机低秩矩阵逼近

随机 SVD 已成为有效计算矩阵低秩近似的极其成功的方法。特别是 Halko、Martinsson 和 Tropp 的论文 (SIREV 2011) 包含了大量分析，使其成为一种非常流行的方法。对于密集矩阵，$m\times n$ 矩阵的 rank-$r$ 近似的典型复杂度为 $O(mn\log n+(m+n)r^2)$。经典的 Nystr{\"o}m 方法要快得多，但仅适用于半正定矩阵。这项工作研究了适用于一般矩阵的 Nystr{\"o}m 方法的推广，并表明 (i) 它具有近- 与竞争方法相当的最佳近似质量，（ii）计算成本是接近最优的 $O(mn\log n+r^3)$ 密集矩阵，具有小的隐藏常数，以及（iii）至关重要，尽管存在病态伪逆，它可以以数值稳定的方式实现。数值实验表明，广义 Nystr{\"o}m 可以显着优于最先进的方法，特别是当 $r\gg 1$ 时，实现高达 10 倍的加速。该方法也非常适合更新并更新矩阵。