A Weighted Randomized Kaczmarz Method for Solving Linear Systems

TL;DR

This paper introduces a weighted randomized Kaczmarz method that accelerates convergence for solving linear systems by selecting equations based on their residuals, and demonstrates empirical benefits including approximation of singular vectors.

Contribution

The paper proposes a novel weighted selection strategy for the Kaczmarz method, improving convergence rates and providing insights into its relation to maximal correction methods.

Findings

Weighted selection speeds up convergence.

Method approximates small singular vectors.

De-randomization as p approaches infinity.

Abstract

The Kaczmarz method for solving a linear system $Ax = b$ interprets such a system as a collection of equations $\left\langle a_i, x\right\rangle = b_i$, where $a_i$ is the $i-$th row of $A$, then picks such an equation and corrects $x_{k+1} = x_k + \lambda a_i$ where $\lambda$ is chosen so that the $i-$th equation is satisfied. Convergence rates are difficult to establish. Assuming the rows to be normalized, $\|a_i\|_{\ell^2}=1$, Strohmer \& Vershynin established that if the order of equations is chosen at random, $\mathbb{E}~ \|x_k - x\|_{\ell^2}$ converges exponentially. We prove that if the $i-$th row is selected with likelihood proportional to $\left|\left\langle a_i, x_k \right\rangle - b_i\right|^{p}$, where $0<p<\infty$, then $\mathbb{E}~\|x_k - x\|_{\ell^2}$ converges faster than the purely random method. As $p \rightarrow \infty$, the method de-randomizes and explains, among other things, why the maximal correction method works well. We empirically observe that the method computes approximations of small singular vectors of $A$ as a byproduct.