On The Structured Manifold Optimization: Reduced-rank and Positive Definite Matrix Estimation

This thesis mainly proposes optimization schemes regarding both types of structured matrices, as well as their applications in statistics. The first structured matrix optimization is proposed under the reduced-rank constraint, and particularly it can be applied for conducting multi-label classification and variable selection simultaneously. The proposed algorithm for optimization is computationally efficient and delivers superior numerical performance in terms of both classification and variable selection accuracy. The asymptotic consistencies are also established to support the advantages of the proposed method. The second structured matrix optimization proposes the estimation of sparse positive definite matrix generated by a generic coordinate descent (CD) algorithm, and particularly it can be applied to the estimation of the high-dimensional covariance matrix and inverse covariance or precision matrix with variable selection.