Low-Rank Matrix Perturbation Analysis And Estimation For Two Classes Of Sparse Covariance Matrices

Matrix perturbation analysis and covariance matrix estimation play impor-tant roles in statistics,which are widely used in low-rank matrix denoising,cluster-ing,matrix calculation,principal component analysis,canonical correlation anal-ysis and etc.Chapter 2 and 3 of this dissertation study perturbation estimation for low-rank matrices,while Chapter 4 discusses the optimal estimation for two classes of covariance matrices based on missing data.In Chapter 2,we first provide a lower bound estimation for the singular vectors of low-rank matrices under maximum norm,which shows the optimality of the upper bound estimation of Fan et al.（J.Q.Fan,W.C.Wang and Y.Q.Zhong.Journal of Machine Learning Research.2018,18（207）:1-42）.Then it is discussed that the optimal estimation of matrix itself and its singular subspaces under Ky Fan norm.Chapter 3 studies the lower bound estimation for the packing number of a sub-unitary matrix ball under unitarily invariant norm sinΘdistance.Our result extends Cai-Ma-Wu’s theorem（T.T.Cai,Z.M.Ma and Y.H.Wu.The Annals of Statistics.2013,41:3074-3110）from the Frobenius norm metric to unitarily invariant norm metrics.Motivated by the work of Cai,Zhang and Zhou,we investigate the optimal estimation for two kinds of covariance matrices of sub-Gaussian random vector with missing data in Chapter 4.It is firstly considered that under matrix l₁norm and Frobenius norm,the optimal estimation of the sparse covariance matrix set defined by weak l_qball.Then under the spectral norm,we give an upper bound estimation for the banded covariance matrix set with additive sub-Gaussian noises.