Interlacing Polynomial Method for Matrix Approximation via Generalized Column and Row Selection (2312.01715v2)

Abstract: This paper delves into the spectral norm aspect of the Generalized Column and Row Subset Selection (GCRSS) problem. Given a target matrix $\mathbf{A}\in \mathbb{R}^{n\times d}$, the objective of GCRSS is to select a column submatrix $\mathbf{B}{:,S}\in\mathbb{R}^{n\times k}$ from the source matrix $\mathbf{B}\in\mathbb{R}^{n\times d_B}$ and a row submatrix $\mathbf{C}{R,:}\in\mathbb{R}^{r\times d}$ from the source matrix $\mathbf{C}\in\mathbb{R}^{n_C\times d}$, such that the residual matrix $(\mathbf{I}n-\mathbf{B}{:,S}\mathbf{B}{:,S}^{\dagger})\mathbf{A}(\mathbf{I}_d-\mathbf{C}{R,:}^{\dagger} \mathbf{C}{R,:})$ has a small spectral norm. By employing the method of interlacing polynomials, we show that the smallest possible spectral norm of a residual matrix can be bounded by the largest root of a related expected characteristic polynomial. A deterministic polynomial time algorithm is provided for the spectral norm case of the GCRSS problem. We next focus on two specific GCRSS scenarios: the Generalized Column Subset Selection (GCSS) problem ($r=0$), and the submatrix selection problem ($\mathbf{B}=\mathbf{C}=\mathbf{I}_d$). In the GCSS scenario, we connect the expected characteristic polynomials to the convolution of multi-affine polynomials, leading to the derivation of the first provable reconstruction bound on the spectral norm of a residual matrix. In the submatrix selection scenario, we show that for any sufficiently small $\varepsilon>0$ and any square matrix $\mathbf{A}\in\mathbb{R}^{d\times d}$, there exist two subsets $S\subset [d]$ and $R\subset [d]$ of sizes $O(d\cdot \varepsilon^2)$ such that $\Vert\mathbf{A}{S,R}\Vert_2\leq \varepsilon\cdot \Vert\mathbf{A}\Vert_2$.