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，用$r$列实现的最佳列近似误差，至多比截断奇异值分解的误差大$\sqrt{r+1}$倍。在实践中达到这一界限，要么需要昂贵的随机体采样，要么至少执行$r$次奇异值分解。本文将证明，仅需单次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$O(Nr^2)$次运算，在大小为$N$的$r$行中找到一个高度非退化的子矩阵，该子矩阵的性质与最大体子矩阵基本相同。