Research On Optimization Algorithms For Sparse Estimation Of High-Dimensional Correlation Matrix

Covariance matrix estimation or correlation matrix estimation is a classic problem in the fields of statistics.This problem is widely used in the fields of high dimensional data analysis,such as economy,finance,social network,gene sequencing and so on.The combination of statistical analysis and optimization algorithms are the 'latest research topics at the present,and the numerical optimization algorithms are widely implemented to solve optimization models in statistics.Due to the simple iteration and low storage capacity,the alternating direction method of multiplier(ADMM)and the accelerated proximal gradient algorithm(APG)are efficient for solving separable convex optimization problems.This thesis focuses on the applications of ADMM and APG in the sparse estimation of high-dimensional correlation matrix.The convergence of the algorithm is listed,and the numerical experiments on simulated data are also included.In the first chapter,some frequently used basic concepts in statistics and optimization are summarized.The problem of estimating of covariance matrix and correlation matrix as well as the corresponding algorithms are briefly reviewed.Some symbols and concepts used in this thesis are also listed.In the second chapter,we briefly review the classical ADMM,generalized ADMM,symmetric ADMM and APG as well as the corresponding convergence theorem.Finally,we also state the main motivation and contribution of this thesis.In the third chapter,based on the correlation matrix estimation model proposed by Cui,Leng&Sun(2016)and Liu,Wang&Zhao(2014),we propose a new estimation model where the explicit constraints on the maximum and minimum eigenvalues are contained.The estimation model is then solved by using several types of ADMM,and the convergence of the algorithm is reported.The effectiveness of each algorithm and the superiority of the proposed model are verified by numerical experiments.In the forth chapter,based on the new correlation matrix estimation model proposed in the previous chapter,we deduce its corresponding dual problem.Then,we employ the well-known APG method to solve the dual model where a smooth term and a non-smooth term are contained.Finally,the convergence rate theorem of the algorithm is given.The effectiveness of the algorithm and the superiority of the model are tested using simulated data.In the fifth chapter,we summarize our thesis and list some research topics for further study.