Natural Thresholding Algorithms for Signal Recovery With Sparsity

The algorithms based on the technique of optimal k -thresholding (OT) were recently proposed for signal recovery, and they are very different from the traditional family of hard thresholding methods. However, the computational cost for OT-based algorithms remains high at the current stage of their development. This stimulates the development of the so-called natural thresholding (NT) algorithm and its variants in this paper. The family of NT algorithms is developed through the first-order approximation of the so-called regularized optimal k -thresholding model, and thus the computational cost for this family of algorithms is significantly lower than that of the OT-based algorithms. The guaranteed performance of NT-type algorithms for signal recovery from noisy measurements is shown under the restricted isometry property and concavity of the objective function of regularized optimal k -thresholding model. Empirical results indicate that the NT-type algorithms are robust and very comparable to several mainstream algorithms for sparse signal recovery.

Under the sparsity assumption, the problem of signal recovery from nonadaptive linear measurements can be formulated as a sparse optimization problem which may take different forms, depending on such factors as recovery environment, signal structure, and the prior information available for the signal [1], [2]. A fundamental mathematical model for sparsity-based signal recovery can be described as follows. Let  A  be an  m×n  sensing matrix with  m≪n,  and let  y:=Ax+ν∈Rm  be the measurements for the signal  x∈Rn , where  ν∈Rm  are the measurement errors. When  x  is  k -sparse or  k -compressible, the recovery of  x  can be formulated as the following sparsity-constrained optimization (SCO) problem: