Interlacing Polynomial Method for Matrix Approximation via Generalized Column and Row Selection

This paper delves into the spectral norm aspect of the Generalized Column and Row Subset Selection (GCRSS) problem. Given a target matrix \textbf{A}\in \mathbb {R}^{n\times d}$\textbf{A}\in \mathbb {R}^{n\times d}$, the objective of GCRSS is to select a column submatrix \textbf{B}_{:,S}\in \mathbb {R}^{n\times k}$\textbf{B}_{:,S}\in \mathbb {R}^{n\times k}$ from the source matrix \textbf{B}\in \mathbb {R}^{n\times d_B}$\textbf{B}\in \mathbb {R}^{n\times d_B}$ and a row submatrix \textbf{C}_{R,:}\in \mathbb {R}^{r\times d}$\textbf{C}_{R,:}\in \mathbb {R}^{r\times d}$ from the source matrix \textbf{C}\in \mathbb {R}^{n_C\times d}$\textbf{C}\in \mathbb {R}^{n_C\times d}$, such that the residual matrix (\textbf{I}_n-\textbf{B}_{:,S}\textbf{B}_{:,S}^{\dagger })\textbf{A}(\textbf{I}_d-\textbf{C}_{R,:}^{\dagger } \textbf{C}_{R,:})$(\textbf{I}_n-\textbf{B}_{:,S}\textbf{B}_{:,S}^{\dagger })\textbf{A}(\textbf{I}_d-\textbf{C}_{R,:}^{\dagger } \textbf{C}_{R,:})$ 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 apply our results to two specific GCRSS scenarios, one where r=0$r=0$, simplifying the problem to the Generalized Column Subset Selection (GCSS) problem, and the other where \textbf{B}=\textbf{C}=\textbf{I}_d$\textbf{B}=\textbf{C}=\textbf{I}_d$, reducing the problem to the submatrix selection problem. 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$\varepsilon >0$ and any square matrix \textbf{A}\in \mathbb {R}^{d\times d}$\textbf{A}\in \mathbb {R}^{d\times d}$, there exist two subsets S\subset [d]$S\subset [d]$ and R\subset [d]$R\subset [d]$ of sizes O(d\cdot \varepsilon ^2)$O(d\cdot \varepsilon ^2)$ such that \Vert \textbf{A}_{S,R}\Vert _2\le \varepsilon \cdot \Vert \textbf{A}\Vert _2$\Vert \textbf{A}_{S,R}\Vert _2\le \varepsilon \cdot \Vert \textbf{A}\Vert _2$. Unlike previous studies that have produced comparable results for very special cases where the matrix is either a zero-diagonal or a positive semidefinite matrix, our results apply universally to any square matrix \textbf{A}$\textbf{A}$.