Asymptotically Sharp Upper Bound for the Column Subset Selection Problem

This paper investigates the spectral norm version of the column subset selection problem. Given a matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ and a positive integer $k\leq\text{rank}(\mathbf{A})$, the objective is to select exactly $k$ columns of $\mathbf{A}$ that minimize the spectral norm of the residual matrix after projecting $\mathbf{A}$ onto the space spanned by the selected columns. We use the method of interlacing polynomials introduced by Marcus-Spielman-Srivastava to derive an asymptotically sharp upper bound on the minimal approximation error, and propose a deterministic polynomial-time algorithm that achieves this error bound (up to a computational error). Furthermore, we extend our result to a column partition problem in which the columns of $\mathbf{A}$ can be partitioned into $r\geq 2$ subsets such that $\mathbf{A}$ can be well approximated by subsets from various groups. We show that the machinery of interlacing polynomials also works in this context, and establish a connection between the relevant expected characteristic polynomials and the $r$-characteristic polynomials introduced by Ravichandran and Leake. As a consequence, we prove that the columns of a rank-$d$ matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ can be partitioned into $r$ subsets $S_1,\ldots S_r$, such that the column space of $\mathbf{A}$ can be well approximated by the span of the columns in the complement of $S_i$ for each $1\leq i\leq r$.