Efficient Approximation of ℓp-Lewis Weights Using Low-Precision Leverage Score Computations

Core Concepts

This paper presents a simple method to compute two-sided ε-approximations of the ℓp-Lewis weights of an n × d matrix using only poly(d, 1/ε) many poly(1/d, ε)-approximate leverage score computations.

Abstract

The paper focuses on efficiently computing approximate ℓp-Lewis weights, a generalization of leverage scores that measure the importance of rows with respect to the ℓp norm.
Key highlights:

Lewis weights lack an explicit description and are more challenging to compute than leverage scores, especially for p ≥ 4.
Prior work has established algorithms for computing ε-estimates of Lewis weights, but they require precision inversely polynomial in n, which can be prohibitive.
The authors show that by running a simple "fixed point iteration" on a one-sided ε-approximate Lewis weight, one can obtain a two-sided O(εpd)-approximation using only poly(d, 1/ε) many poly(1/d, ε)-approximate leverage score computations.
This effectively also yields a O(p^3 d^3/2 ε)-estimate of the true Lewis weights.
The authors provide an algorithm that combines their result with a previous one-sided approximation algorithm, allowing for the computation of two-sided Lewis weight approximations from low-precision leverage scores.

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Key Insights Distilled From

On computing approximate Lewis weights

by Simon Apers,... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02881.pdf

Deeper Inquiries

How can the techniques presented in this paper be extended to obtain even stronger notions of Lewis weight approximations, such as multiplicative ε-estimates

To obtain even stronger notions of Lewis weight approximations, such as multiplicative ε-estimates, the techniques presented in the paper can be extended by refining the iterative schemes used for computing leverage scores. By incorporating more sophisticated iterative methods or refining the approximation algorithms for leverage scores, it is possible to enhance the accuracy and precision of the computed Lewis weights. Additionally, exploring advanced mathematical techniques or optimization strategies could lead to more robust and accurate multiplicative estimates of Lewis weights.

What are the implications of the authors' results for applications that rely on Lewis weights, such as ℓp-embedding, ℓp-regression, and interior point methods

The results presented in the paper have significant implications for various applications that rely on Lewis weights, such as ℓp-embedding, ℓp-regression, and interior point methods. By providing efficient algorithms for computing approximate Lewis weights with controlled accuracy, the authors' findings enable improved performance and reliability in these applications. For ℓp-embedding and ℓp-regression problems, accurate Lewis weights are crucial for optimizing the solutions and achieving better outcomes. In interior point methods, precise Lewis weights play a vital role in determining the convergence and efficiency of the optimization process. Therefore, the algorithms proposed in the paper can enhance the effectiveness and applicability of these applications.

Can the ideas in this paper be applied to other matrix importance measures beyond Lewis weights, such as ridge leverage scores or sampling probabilities for randomized linear algebra

The ideas presented in the paper can potentially be applied to other matrix importance measures beyond Lewis weights, such as ridge leverage scores or sampling probabilities for randomized linear algebra. By adapting the algorithms and techniques developed for computing Lewis weights, it may be possible to derive similar approximation methods for these alternative matrix importance measures. This extension could involve modifying the iterative schemes or approximation algorithms to suit the specific properties and requirements of ridge leverage scores or sampling probabilities. By leveraging the foundational concepts and methodologies introduced in the paper, it is feasible to explore the application of these ideas to a broader range of matrix importance measures in various computational contexts.