  Robust compressive sensing(CS) reconstruction has become an attractive
research topic in recent years. Robust CS aims to reconstruct the sparse
signals under non-Gaussian(i.e. heavy tailed) noises where traditional CS
reconstruction algorithms may perform very poorly due to utilizing $l_2$ norm
of the residual vector in optimization. Most of existing robust CS
reconstruction algorithms are based on greedy pursuit method or convex
relaxation approach. Recently, the adaptive filtering framework has been
introduced to deal with the CS reconstruction, which shows desirable
performance in both efficiency and reconstruction performance under Gaussian
noise. In this paper, we propose an adaptive filtering based robust CS
reconstruction algorithm, called $l_0$ regularized maximum correntropy
criterion($l_0$-MCC) algorithm, which combines the adaptive filtering framework
and maximum correntropy criterion(MCC). MCC has recently been successfully used
in adaptive filtering due to its robustness to impulsive non-Gaussian noises
and low computational complexity. We analyze theoretically the stability of the
proposed $l_0$-MCC algorithm. A mini-batch based $l_0$-MCC(MB-$l_0$-MCC)
algorithm is further developed to speed up the convergence. Comparison with
existing robust CS reconstruction algorithms is conducted via simulations,
showing that the proposed $l_0$-MCC and MB-$l_0$-MCC can achieve significantly
better performance than other algorithms.
