Research On Greedy Algorithm Based On Bi-level Sparse Structure

Sparse optimization has become the forefront topic of current research.The main idea of sparse optimization is to use sparse structure to extract and reconstruct high-dimensional signals from a small amount of data information,which has been widely used in signal processing,machine learning and other fields.Structural sparse optimization is an important problem in the field of sparse optimization.Its main purpose is to use the specific structure of the problem to improve the ability of sparse optimization and it has a wide range of applications in the engineering field.Peng study the relationship between DNA and RNA transcription regulation discovering that problem not only have sparse structures outside the group,but also have sparse structures within the group.From this,a bi-level sparse optimization is derived,which is widely used in genetic engineering,spectral analysis and other fields.The greedy algorithm has the advantages of simple thought,convenient operation,and fast calculation.Therefore,it is a common method in the field of information engineering.Orthogonal matching pursuit algorithm(OMP)is more classic among greedy algorithms.Because of its advantages such as lower complexity and faster convergence speed,it has been widely loved by researchers in recent years.However,this algorithm is a forward greedy algorithm.There is no way to remove the wrong set of indicators in the previous iteration steps.Therefore,Zhang Tong and others proposed an adaptive forward and backward greedy algorithm(FOBA)and gave its theoretical properties,laying a theoretical foundation for future greedy algorithm research.Based on this,this paper introduces these two algorithms to the bi-level sparse optimization problem,and proposes bi-level othogonal matching pursuit algorithm(BLOMP)and bi-level forward and backward greedy algorithm(BLFOBA),and study the theoretical properties of the BLOMP algorithm and the effects of numerical simulation experiments.The main work and contributions of this article are as follows:First of all,this paper proposes a BLOMP algorithm to solve such inter-state problems based on some practical problems with a bi-level sparse structure,and generalizes the OMP algorithm to the bi-level sparse optimization problem.Then,this paper studies the theoretical nature of the BLOMP algorithm.The model is divided into two cases: no noise and noisy.In the absence of noise,it is obtained that this algorithm can select the correct support set under the condition of accurate recovery,that is,the signal can be accurately recovered.In the presence of noise,the conditions for the algorithm to select the correct support set are given.At the same time,the convergence properties when the noise is Gaussian are studied,and the upper bound of the error between the model solution and the real solution is given.This lays the theoretical foundation for the application of the algorithm in the engineering field.Thirdly,this paper generalizes the improved FOBA algorithm of the forward greedy algorithm to the bi-level sparse optimization problem,and proposes the BLFOBA algorithm to overcome the disadvantage of the forward greedy algorithm that cannot select the wrong support set for deletion.Last but not least,this paper simulates the numerical experiments of the two algorithms,and compares the algorithm designed in this paper with some classic greedy algorithms,such as OMP,FOBA algorithm,and group orthogonal matching pursuit algorithm(GOMP).Experiments show that under the premise of using the bi-level sparse structure of the data,the performance of the designed algorithms in this paper is better than these algorithms.