The sparse Kaczmarz method with surrogate hyperplane for the regularized basis pursuit problem

TL;DR

This paper introduces a surrogate hyperplane variant of the Sparse Kaczmarz method for the regularized basis pursuit problem, providing convergence analysis and demonstrating improved efficiency through numerical experiments.

Contribution

It proposes a new surrogate hyperplane approach for the Sparse Kaczmarz method, with proven convergence and detailed analysis of linear convergence rates.

Findings

The surrogate hyperplane Sparse Kaczmarz method converges linearly.

Numerical experiments show improved efficiency over existing methods.

Theoretical convergence rates are rigorously established.

Abstract

The Sparse Kaczmarz method is a famous and widely used iterative method for solving the regularized basis pursuit problem. A general scheme of the surrogate hyperplane sparse Kaczmarz method is proposed. In particular, a class of residual-based surrogate hyperplane sparse Kaczmarz method is introduced and the implementations are well discussed. Their convergence theories are proved and the linear convergence rates are studied and compared in details. Numerical experiments verify the efficiency of the proposed methods.