  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.
