Close to optimal column approximation using a single SVD

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. In this paper it will be shown that this optimal column approximation bound can be reached with only a single SVD. Moreover, it can be approximately reached in just a single approximate truncated SVD in comparison with r full SVDs in the original approach, thus making close to optimal column subset selection not more complex than the commonly used matrix approximation based on random projections. As a corollary, it will be shown how to find a highly nondegenerate submatrix in r rows of size N in just $O\left(N{r}^2\right)$ operations, which mostly has the same properties as the maximum volume submatrix.