  Sparse signals can be possibly reconstructed by an algorithm which merges a
traditional nonlinear optimization method and a certain thresholding technique.
Different from existing thresholding methods, a novel thresholding technique
referred to as the optimal $k$-thresholding was recently proposed by Zhao [SIAM
J Optim, 30(1), pp. 31-55, 2020]. This technique simultaneously performs the
minimization of an error metric for the problem and thresholding of the
iterates generated by the classic gradient method. In this paper, we propose
the so-called Newton-type optimal $k$-thresholding (NTOT) algorithm which is
motivated by the appreciable performance of both Newton-type methods and the
optimal $k$-thresholding technique for signal recovery. The guaranteed
performance (including convergence) of the proposed algorithms are shown in
terms of suitable choices of the algorithmic parameters and the restricted
isometry property (RIP) of the sensing matrix which has been widely used in the
analysis of compressive sensing algorithms. The simulation results based on
synthetic signals indicate that the proposed algorithms are stable and
efficient for signal recovery.
