On computing approximate Lewis weights

In this note we provide and analyze a simple method that given an n × d matrix, outputs approximate ℓp-Lewis weights, a natural measure of the importance of the rows with respect to the ℓp norm, for p ≥ 2. More precisely, we provide a simple post-processing procedure that turns natural one-sided approximate ℓp-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 ℓp-Lewis weights of an n × d-matrix using poly(d,p) approximate leverage score computations. While efficient high-accuracy algorithms for approximating ℓp-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.