Subset selection for matrices by column exchange

The paper considers the problem of finding a submatrix $X_{\mathcal{S}} \in \mathbb{R}^{m \times k}$ in a matrix $X \in \mathbb{R}^{m \times n}$, such that the spectral or Frobenius norm of $X_{\mathcal{S}}^{\dag} X$ is limited, which guarantees it provides a good representation of the whole matrix. Such bounds can be reached by applying greedy algorithms, maximizing the submatrix volume. We suggest a modification of a greedy volume maximization, which performs column exchanges asymptotically faster for $n \gg m$ than the known alternatives, while guaranteeing the same bounds on $X_{\mathcal{S}}^{\dag} X$. In addition, we prove a new upper bound on the number of required exchanges, which is applicable to the new algorithm as well as to other greedy volume maximization algorithms.