The best column approximation in the Frobenius norm with $r$ columns has an error at most $\sqrt{r+1}$ times larger than the truncated singular value decomposition. Reaching this bound in practice involves either expensive random volume sampling or at least $r$ executions of singular value decomposition. In this paper it will be shown that the same column approximation bound can be reached with only a single SVD (which can also be replaced with approximate SVD). As a corollary, it will be shown how to find a highly nondegenerate submatrix in $r$ rows of size $N$ in just $O(Nr^2)$ operations, which mostly has the same properties as the maximum volume submatrix.

翻译：在Frobenius范数下，使用$r$列的最佳列近似误差最多比截断奇异值分解大$\sqrt{r+1}$倍。实际中达到这一界限需要昂贵的随机体积采样或至少执行$r$次奇异值分解。本文表明，仅需一次SVD（也可用近似SVD替代）即可达到相同的列近似界限。作为推论，本文将展示如何仅用$O(Nr^2)$次操作在大小为$N$的$r$行中找到一个高度非退化的子矩阵，该子矩阵与最大体积子矩阵具有几乎相同的性质。