On Computing Approximate Lewis Weights

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