In this note we provide and analyze a simple method that given an $n \times d$ matrix, outputs approximate $\ell_p$-Lewis weights, a natural measure of the importance of the rows with respect to the $\ell_p$ norm, for $p \geq 2$. More precisely, we provide a simple post-processing procedure that turns natural one-sided approximate $\ell_p$-Lewis weights into two-sided approximations. When combined with a simple one-sided approximation algorithm presented by Lee (PhD thesis, `16) this yields an algorithm for computing two-sided approximations of the $\ell_p$-Lewis weights of an $n \times d$-matrix using $\mathrm{poly}(d,p)$ approximate leverage score computations. While efficient high-accuracy algorithms for approximating $\ell_p$-Lewis had been established previously by Fazel, Lee, Padmanabhan and Sidford (SODA `22), the simple structure and approximation tolerance of our algorithm may make it of use for different applications.

翻译：本文提出并分析了一种简单方法，对于给定的 $n \times d$ 矩阵，该方法能输出 $\ell_p$-Lewis 权重的近似值——即针对 $\ell_p$ 范数的行重要性自然度量（适用于 $p \geq 2$）。具体而言，我们设计了一种简单的后处理流程，可将自然单侧近似的 $\ell_p$-Lewis 权重转化为双侧近似。结合 Lee（博士论文，2016）提出的简单单侧近似算法，该方法能通过 $\mathrm{poly}(d,p)$ 次近似杠杆值计算，得到 $n \times d$ 矩阵 $\ell_p$-Lewis 权重的双侧近似。尽管此前 Fazel、Lee、Padmanabhan 和 Sidford（SODA 2022）已建立高效的 $\ell_p$-Lewis 权重高精度近似算法，但本算法的简单结构及近似容差特性可能使其适用于不同场景。