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Both least mean square (LMS) and least mean fourth (LMF) are popular adaptive algorithms with application to adaptive channel estimation. Because the wireless channel vector is often sparse, sparse LMS-based approaches have been proposed with different sparse penalties, for example, zero-attracting LMS and Lp-norm LMS. However, these proposed methods lead to suboptimal solutions in low signal-to-noise ratio (SNR) region, and the suboptimal solutions are caused by LMS-based algorithms that are sensitive to the scaling of input signal and strong noise. Comparatively, LMF can achieve better solution in low SNR region. However, LMF cannot exploit the sparse information because the algorithm depends only on its adaptive updating error but neglects the inherent sparse channel structure. In this paper, we propose several sparse LMF algorithms with different sparse penalties to achieve better solution in low SNR region and take the advantage of channel sparsity at the same time. The contribution of this paper is briefly summarized as follows: (1) construct the cost functions of the LMF algorithm with different sparse penalties; (2) derive their lower bounds; and (3) provide experiment results to show the performance advantage of the propose method in low SNR region. [ABSTRACT FROM AUTHOR]