Approximation To ?₀ Norm And Its Application In Sparse Optimization

Compressed Sensing?CS?has been rapidly growing fields of research in many practical problems of science and technology over the past decade.It not only enriches the research contents of the digital signal processing,but also provides new approaches and ideas for researching in the other fields of expertise and has a wide application future.We mainly focus on the?₀minimization problem by using some classes of sparse approximating functions.We construct two new weights?₁-algorithm and?₂-algorithm by using non-convex and non-smooth approximation functions instead of?₀norm and then considers the relation between a solution of the?₀minimization problem and the least squares solution by utilizing continuously differentiable function.Firstly,in this thesis,we construct two classes of non-convex,non-smooth and Lipschitz sparse approximating functions to the?₀norm by using a unique solution of the Moreau envelope of the?₁norm.Based on these sparse approximating functions,we derive some inequalities about?₀norm and build the crucial link with generalized minimax-concave.Then we deduce two new weights?₁algorithm and?₂algorithm by applying these sparse approximating functions to compressed sensing and give the proof for the boundedness of the iterative sequence generated by these two algorithms.In addition,we explain why the solution to reweighted?₁minimization is better than one of the basis pursuit problem by building a constrained mixed optimization problem.Secondly,we study a special ?₁/?₂minimization problem.Since the objective function of this problem is a non-convex,non-smooth and non-Lipschitz function,we mainly consider the algorithm solving this problem.We start with simple iteratively reweighted?₁algorithm.To prove the convergence of this reweighted?₁algorithm,we construct a truncated iterative soft thresholding?TIST?algorithm based on ISTA and further prove the convergence of TIST.Thirdly,we study the relationship between a solution of the cardinality minimization problem and the least squares solution by the continuously differentiable function of the?₀norm.Based on the smoothed?₀norm,we obtain an approximate problem of the?₀minimization problem under the condition that sensing matrix satisfies the unique representation property.Subsequently,we decompose the approximate problem into the basis pursuit problem which has distinct observed data and the unconstrained one by using a simple conversion techniques.Based on the above processing,we can obtain a sparse solution to the?₀minimization by solving these two subproblems.Finally,we present some numerical experiments to verify the feasibility and effec-tiveness of the constructed method in the last subsection of each chapter.Since some test comparisons of the other state-of-the-art CS solvers were done,we compare with some commonly used algorithms for signal recovery success rate in the application of signal reconstruction and image denoising.