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.

翻译：随机稀疏Kaczmarz方法旨在求解线性系统$Ax=b$的稀疏解，其以正比于$\|a_{i}\|_2^2$的概率选取第$i$个投影超平面，其中$a_{i}^T$为$A$的第$i$行。本文提出一种加权随机稀疏Kaczmarz方法，该方法以正比于$\lvert\langle a_{i},x_{k}\rangle-b_{i}\rvert^p$的概率选取第$i$个投影超平面（$0<p<\infty$），以实现可能的加速。该方法通过参数$p$桥接了随机Kaczmarz方法与贪婪Kaczmarz方法。理论上，我们证明其在无噪声和含噪声情况下关于Bregman距离的期望线性收敛速率，且该速率至少与随机稀疏Kaczmarz方法相当。通过一组数值实验验证了所提方法的优越性。