Subset selection for matrices by column exchange

TL;DR

This paper introduces a faster greedy algorithm for selecting submatrices that effectively represent the original matrix, with proven bounds on the number of exchanges needed.

Contribution

It proposes a modified greedy volume maximization algorithm with improved exchange efficiency for large matrices, maintaining theoretical bounds.

Findings

The new algorithm performs column exchanges asymptotically faster for large n.

It guarantees the same spectral or Frobenius norm bounds as existing methods.

A new upper bound on the number of exchanges is established.

Abstract

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.