Iterative Soft Thresholding Algorithm And Its Accelerated Versions For Elastic-net Regularization

Recently,sparsity regularization,as an effective regularization technique,has been widely concerned.However,the classical l1-sparsity regularization fails to have good stability for ill-posed problems with large condition numbers.While elastic-net regularization has better stability,and it is more suitable for inverse problems with large condition numbers.Elastic-net regularization has become a new research direction in the field of inverse problems.At present,the two numerical algorithms for elastic-net regularization are mainly based on Active set.The aim of this paper is to construct iterative soft thresholding algorithm and its accelerated versions for elastic-net regularization.Firstly,a new iterative soft thresholding algorithm is proposed,and the generalized conditional gradient algorithm is extended to elastic-net regularization.As a numerical algorithm for sparsity regularization,iterative soft thresholding algorithm has a simple formulation and it can easily be implemented.Its convergence is proved.In addition,the effectiveness of the algorithm is testified by numerical experiments.Numerical experiments indicate that,in the presence of noise,elastic-net regularization has better performance than the classical l1-sparsity regularization.Even though iterative soft thresholding algorithm can easily be implemented,in general,it can be arbitrarily slow and it is computationally intensive.Therefore,the two accelerated algorithms are presented,and the projected gradient method is extended to elastic-net regularization.One is the projected gradient algorithm based on the generalized conditional gradient method,and another is based on the surrogate function approach.Subsequently,the weak convergence and strong convergence of the projection gradient algorithm based on the surrogate function approach are proved.In addition,a strategy to determine the radius R of the l1-ball constraint is proposed in the projected gradient method by Morozov's discrepancy principle.Numerical experiments indicate that the two projected gradient algorithms have faster convergence rates compared to the iterative soft thresholding algorithm.Hence,the projected gradient algorithm is adequate for solving large-scale ill-posed problems.