New Statistical Learning Methods for Multiple High Dimensional Datasets

In this dissertation, we design several statistical learning methods for analyzing multiple high-dimensional datasets. Our focus is on multiple response regression and inverse covariance matrix estimation.;Multivariate regression is a common statistical tool for practical problems. Many multivariate regression techniques are designed for univariate response cases. For problems with multiple response variables available, one common approach is to apply the univariate response regression technique separately on each response variable. Although it is simple and popular, the univariate response approach ignores the joint information among response variables. We propose several methods for utilizing joint information among response variables in a penalized likelihood framework. The proposed methods provide sparse estimators for the conditional inverse covariance matrix of response vector given explanatory variables as well as sparse estimators of regression coefficient matrix.;Estimation of inverse covariance matrices is important in various areas of statistical analysis. The task of estimating multiple inverse covariance matrices sharing some common structure is considered. In this case, estimating each matrix separately can be suboptimal as it ignores potential common structure. We propose a new approach to parameterize each inverse covariance matrix as a sum of common and unique components and jointly estimate multiple inverse covariance matrices in a constrained L1 minimization framework.;Theoretical properties of the new methods are explored. Simulated examples and applications to a glioblastoma multiforme cancer data demonstrate competitive performance of the proposed methods.