Probabilistic Constrained Optimization Model In Compressed Sensing And Its Smooth Approximation

Compressed sensing is a sampling theory newly emerged in recent years.It is also called compressed sampling.It is widely used in wireless communication,microwave imaging,and pattern recognition.The most important part of compressed sensing theory is to select an appropriate observation matrix.A more accurate image can be reconstructed with a small amount of observations.The representative solution method is convex relaxation method,including BP algorithm,gradient projection method,Bregman iteration method,etc.It is very difficult to judgment the restricted isometry property(RIP)and non-interfering property of observation matrix.The observation signal is easily polluted by noise in the actual measurement process,which causes errors in the reconstruction process.In view of this,this thesis proposes the probabilistic constrained optimization model in compressed sensing and discusses smooth approximation method.The main research contents are as follows:Chapter 1 introduces the background of compressive sensing problem and the research status of the probabilistic constrained optimization problem.Related preliminary knowledge is presented.Chapter 2 builds a probabilistic constrained optimization model for compressive sensing problems.According to the randomness of the observation matrix,signal recombination problem with noisy is reconstructed as l1- norm minimization problem with probabilistic constraints.The properties of l1- norm and probability constrained function are discussed.In chapter 3,a smooth approximation problem for l1- norm minimization with probability constraints is established based on non-differentiability of probability functions.Firstly,a smooth D.C.approximation function (?)(z,t)of the characteristic function 1(0,+?)(z) is defined,and the properties of the function (?)(z,t)are discussed.Secondly,The smooth D.C.approximation problem ((?)) is constructed based on the function.The equivalence is proved under certain conditions.Finally,e-approximate problem (P?) of the smooth approximation problem ((?)) is established and the convergence analysis is performed.Chapter 4 discusses sequential convex approximation(SCA)for solving smooth D.C.approximation problem(P?).Firstly,sequential convex approximation algorithm is given.Secondly,the initial solution of the algorithm is discussed.Finally,a sample averageapproximation(SAA)method for solving subproblems is presented,and convergence of algorithm(SAA)is analyzed.