ASIHT Algorithm For LAD Problems With Sparsity Constraints

As a new standard sampling theory,compressed sensing is mainly used for signal recovery based on sparse constraints,and is widely used in signal processing,medical imaging and other fields.In the context of compressed sensing,this thesis aims to solve the least absolute deviations problem with sparse constraints,because of its nonsmooth objective function and nonconvex constraint space,the solution of this problem is somewhat challenging.The proposal of the adaptive iterative hard threshold algorithm has effectively solved this problem,however,this algorithm is prone to divergence in practical applications,based on this phenomenon,this thesis will improve the algorithm and its convergence theorem based on this phenomenon.Firstly,we analyze the shortcomings of the adaptive iterative hard threshold algorithm by combining two groups of experimental phenomena.Aiming at the problem that signal reconstruction error decline cannot be guaranteed in the algorithm,we find the essential cause of divergence phenomenon,and obtain the defect of invariability of step coefficient.Based on this,we optimize and improve the iterative step size,an adaptive step iterated hard threshold algorithm is proposed,the step coefficient of the algorithm can enter the convergence range adaptively during iteration to ensure the convergence of the algorithm.Secondly,we provide the convergence proof of adaptive step iterated hard threshold algorithm.When the step coefficient enters the convergence interval,it will converge at a linear rate.In addition,a more accurate and broader range of parameters is given to reveal the correspondence between different parameters.At the same time,the optimal step size coefficient value which can make the algorithm reach the fastest speed is theoretically obtained,which provides a clearer convergence criterion for the algorithm.Finally,we test the performance of adaptive step iterated hard threshold algorithm through three sets of numerical experiments,and verify the effectiveness of the algorithm from three aspects: feasibility,convergence and robustness.