Joint Estimation of Multiple High-Dimensional Precision Matrices with an Application in Genomics Data

• Activity Number: 18
• Type: Topic Contributed
• Date/Time: Sunday, August 4, 2013 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Computing
• Author(s): Jichun Xie*+ and Weidong Liu and Hongzhe Li and Tony Cai
• Companies: Temple University and Shanghai Jiao Tong University and University of Pennsylvania and University of Pennsylvania
• Keywords: Constrained optimization ; Graph recovery ; Second-order cone programming

Abstract

Estimation of high dimensional precision matrix has many important applications. Motivated by the analysis of gene expression data measured in different tissues or disease states, we consider the problem of jointly estimating multiple precision matrices in order to effectively utilize the partially shared graphical structures of the corresponding graphs. The procedure is based on a weighted constrained l_\infty/l_1 minimization, which can be effectively solved by a second-order cone programming. It is shown that under the entry-wise l_\infty norm loss such a joint estimation of precision matrices leads to a faster convergence rate than estimating the precision matrices separately. After thresholding, the joint estimation can have a faster rate than separate estimation under the matrix l_1 norm. We also show that when the collection of the precision matrices have a common support, the proposed procedure leads to the exact graph structure recovery with probability tending to 1. The method is illustrated through an analysis of a mouse liver gene expression data in male and female mice.