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

TL;DR

The paper studies spectral-norm reconstruction in the Generalized Column and Row Subset Selection (GCRSS) problem by linking the worst-case residual to the largest root of an expected polynomial $P_{k,r}(x;\mathbf{A},\mathbf{B},\mathbf{C})$ via the method of interlacing polynomials. It develops a deterministic polynomial-time algorithm that selects column and row subsets to guarantee a bound of the form $\|\mathbf{A}-\mathbf{B}_{:,\widehat{S}}\mathbf{B}_{:,\widehat{S}}^{\dagger}\mathbf{A}\|_2^2 \le \|\mathbf{A}-\mathbf{B}\mathbf{B}^{\dagger}\mathbf{A}\|_2^2 + \varepsilon\|\mathbf{B}\mathbf{B}^{\dagger}\mathbf{A}\|_2^2$ under a controlled parameter $\varepsilon$, with $\delta_k$ and $\alpha$ governing the bounds. The GCSS result yields the first provable reconstruction bound for spectral-norm GCSS via multivariate barrier methods and convolution of multi-affine polynomials, while the submatrix-selection analysis (with $\mathbf{B}=\mathbf{C}=\mathbf{I}_d$) shows existence of small index sets $S,R$ with $|S|,|R|=O(d\varepsilon^2)$ that bound $\|\mathbf{A}_{S,R}\|_2$ by $\varepsilon\|\mathbf{A}\|_2$ using Laguerre-derivative techniques. Collectively, these results provide deterministic guarantees for spectral-norm preservation under structured column/row sampling and connect GCRSS to volume sampling and multivariate polynomial techniques. The framework advances interpretable, provable matrix approximations with explicit sampling guarantees and efficient algorithms.

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 $\Vert\mathbf{A}_{S,R}\Vert_2\leq \varepsilon\cdot \Vert\mathbf{A}\Vert_2$.

Theorems & Definitions (71)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.4
• Remark 1.5