A Weighted Randomized Sparse Kaczmarz Method for Solving Linear Systems

The randomized sparse Kaczmarz method, designed for seeking the sparse solutions of the linear systems $Ax=b$, selects the $i$-th projection hyperplane with likelihood proportional to $\|a_{i}\|_2^2$, where $a_{i}^T$ is $i$-th row of $A$. In this work, we propose a weighted randomized sparse Kaczmarz method, which selects the $i$-th projection hyperplane with probability proportional to $\lvert\langle a_{i},x_{k}\rangle-b_{i}\rvert^p$, where $0<p<\infty$, for possible acceleration. It bridges the randomized Kaczmarz and greedy Kaczmarz by parameter $p$. Theoretically, we show its linear convergence rate in expectation with respect to the Bregman distance in the noiseless and noisy cases, which is at least as efficient as the randomized sparse Kaczmarz method. The superiority of the proposed method is demonstrated via a group of numerical experiments.