  Compressive sensing relies on the sparse prior imposed on the signal of
interest to solve the ill-posed recovery problem in an under-determined linear
system. The objective function used to enforce the sparse prior information
should be both effective and easily optimizable. Motivated by the entropy
concept from information theory, in this paper we propose the generalized
Shannon entropy function and R\'{e}nyi entropy function of the signal as the
sparsity promoting regularizers. Both entropy functions are nonconvex,
non-separable. Their local minimums only occur on the boundaries of the
orthants in the Euclidean space. Compared to other popular objective functions,
minimizing the generalized entropy functions adaptively promotes multiple
high-energy coefficients while suppressing the rest low-energy coefficients.
The corresponding optimization problems can be recasted into a series of
reweighted $l_1$-norm minimization problems and then solved efficiently by
adapting the FISTA. Sparse signal recovery experiments on both the simulated
and real data show the proposed entropy functions minimization approaches
perform better than other popular approaches and achieve state-of-the-art
performances.
