On computing approximate Lewis weights

TL;DR

This work addresses computing approximate $\ell_p$-Lewis weights for an $n\times d$ matrix with $p \ge 2$, introducing a simple post-processing map $\mathcal{T}_p$ that upgrades a one-sided $\varepsilon$-approximation to a two-sided approximation. The core contribution proves that, starting from a one-sided estimate, the transformed vector $\mathcal{T}_p(w)$ yields a two-sided $O(\varepsilon p d)$-approximation to the true Lewis weights, effectively capturing a fixed-point step of the Lewis-weight computation. By combining this with Lee's one-sided algorithm, the paper achieves two-sided estimates using only polynomial-in-$d$ and $p$ approximate leverage-score computations, avoiding the inverse-polynomial dependencies on $n$ required by some prior high-precision methods. The authors further develop a stability-based, multi-pass scheme to obtain multiplicative $\varepsilon$-approximations, with a concrete algorithm and parameter choices that run in $\widetilde{O}(p d/\alpha)$ leverage-score evaluations plus poly$(d,p)$ overhead, making the approach practical for a range of applications including $\ell_p$-regression and convex optimization, and potentially adaptable to quantum-speedup contexts.

Abstract

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.

Theorems & Definitions (16)

• Theorem 1: Low-precision Lewis weights
• Definition 1: Lewis weight approximations
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof