Subset selection for matrices in spectral norm

Abstract: We address the subset selection problem for matrices, where the goal is to select a subset of $k$ columns from a "short-and-fat" matrix $X \in \mathbb{R}^{m \times n}$, such that the pseudoinverse of the sampled submatrix has as small spectral or Frobenius norm as possible. For the NP-hard spectral norm variant, we propose a new deterministic approximation algorithm. Our method refines the potential-based framework of spectral sparsification by specializing it to a single barrier function. This key modification enables direct, unweighted column selection, bypassing the intermediate weighting step required by previous approaches. It also allows for a novel adaptive update strategy for the barrier. This approach yields a new, explicit bound on the approximation quality that improves upon existing guarantees in key parameter regimes, without increasing the asymptotic computational complexity. Furthermore, numerical experiments demonstrate that the proposed method consistently outperforms its direct competitors. A complete C++ implementation is provided to support our findings and facilitate future research.