Computing Lewis weights to high precision using local relative smoothness

2026-06-28

AI Summary

This work addresses the problem of efficiently and accurately computing ℓ_p-Lewis weights (for p ≥ 4) of a matrix, which quantify the importance of its rows. By alternating between primal and dual formulations of the underlying optimization problem and integrating leverage score iteration with a locally relative smooth gradient descent method, the authors propose a novel algorithm that significantly reduces computational overhead while maintaining high accuracy. Specifically, the proposed approach improves the iteration complexity from O(p³ log(m/ε)) to O(p² log(m/ε)), achieving the current best-known bound on the number of iterations required for ε-approximate Lewis weight computation.

Abstract

We provide algorithms that compute $ε$-estimates of the $\ell_p$-Lewis weights of a matrix $A \in \mathbb{R}^{m \times n}$ for $p \geq 4$ using $O(p^2 \log(m/ε))$ rounds of leverage score computation, where $\ell_p$-Lewis weights and leverage scores are both standard measures of row importance. This improves upon the state-of-the-art round complexity of $O(p^3 \log(m/ε))$ due to Fazel, Lee, Padmanabha, and Sidford (2022). We obtain our results by carefully applying a local variant of relatively smooth gradient descent to primal and dual forms of the $\ell_p$-Lewis weight optimization problem and providing tools to convert between different notions of approximate $\ell_p$-Lewis weights.

Problem

• Lewis weights
• leverage scores
• round complexity
• high-precision computation
• row importance

Innovation

• Lewis weights
• relative smoothness
• leverage scores
• gradient descent
• round complexity