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.