Research On High-dimensional Covariance Matrices Estimation Based On Regularization Paths Algorithm And Accelerated Gradient Algorithm

Covariance structure plays a vital role in high-dimensional data analysis,the positivedefinite property of covariance matrix is crucial for the validity of many multivariate statistical procedures.This thesis discuss two algorithmic procedure for covariance matrices estimation in high-dimensional setting.In the first part,we focus on introducing ADMM algorithmic regularization paths for positive-definite large covariance matrices estimation.During the last decade,highdimensional covariance matrices estimation have become more and more popular,while less attention has been attracted to computing regularization paths,or solving the optimization problems over the full range of regularization parameters to obtain a sequence of sparse covariance models.In this part,we aim to use the ADMM Algorithmic Regularization Path for positive-definite covariance matrices to quickly approximate the sequence of sparse covariance models for the purposes of statistical model selection.The numerical results show that our approach not only computationally fast and easy implementation,but also explore the sparse covariance model space at a fine resolution.In the second part,we propose an effective algorithm for the estimation of highdimensional sparse positive-definite covariance matrices.To simultaneously obtain a sparse and positive-definite estimator of large covariance matrices,we consider a positivedefinite constrained 1-penalized minimization problem for estimating sparse large covariance matrices.We use an accelerated gradient method to solve the optimization problem and establish its convergence rate as O(1/k2),where k is the number of iterations.The numerical simulations illustrated our method have an competitive advantage with other methods in computing time,TPR,FPR,convergence rate under Frobenius norm and Spectral norm.