Optimal Threshold Class Algorithms For Sparse Split Feasible Problem

The Sparse Split Feasibility Problem is generated by the combination of the Split Feasibility Problem and the sparsity,which is a very important practical problem.Sparsity means that the vast majority of elements of the signal is 0.Compared to a general signal with the same length,a sparse signal contains less information.Therefore,the sparse signal can be sufficiently compressed to save storage space and reduce the amount of transmission.The sparse split feasibility problem has a wide range of applications,not only in digital signal processing,image processing,compressed sensing,machine learning and other fields,but also in object detection,face recognition,computer vision and other practical problems.Therefore,it is of great practical value to study the feasible and effective algorithms to solve the sparse split feasibility problem.This paper is mainly divided into four chapters.The main structure is shown below:The first part is introduction.The definition,the research situation of the sparse split feasibility problem and the main work of this paper are introduced in this chapter.The second chapter studys a particular form of the sparse split feasibility problem―Compressed Sensing problem,and gives three heavy-ball-based optimal s-thresholding algorithms for solving this problem.These algorithms combine the optimal thresholding operator and the heavy-ball acceleration technology,which can avoid the violent oscillation of the residual compared with the traditional hard thresholding algorithm and accelerate the algorithm through the heavy-ball momentum term.The iterative sequence produced by the algorithms globally converges to a solution of the compressed sensing problem.Finally,numerical examples are given to verify the feasibility and effectiveness of the algorithm.In the third chapter,the optimal thresholding algorithm with the Armijo-like line research for solving the sparse split feasibility problem is presented.This algorithm combines the idea of the optimal thresholding algorithm for solving compressed sensing problem with the armijo-like line research,which avoids the divergence sequence that may be generated by hard thresholding algorithms.The general sparse split feasibility problem can be solved by this algorithm.The convergence of the algorithm is proved.Finally,numerical examples are used to verify the effectiveness and feasibility of the algorithm.Chapter 4 summarizes the research content of the full paper and proposes further research directions.