On computing approximate Lewis weights

In this note we provide and analyze a simple method that given an $n\times d$ matrix, outputs approximate ${\mathrm{\ell }}_p$-Lewis weights, a natural measure of the importance of the rows with respect to the ${\mathrm{\ell }}_p$ norm, for $p\ge 2$. More precisely, we provide a simple post-processing procedure that turns natural one-sided approximate ${\mathrm{\ell }}_p$-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.