Joint Estimation of Multiple High-dimensional Precision Matrices
Tony Cai, Hongzhe Li, Weidong Liu, and Jichun Xie
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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∞/l1 minimization, which can be effectively solved by a second-order cone programming. It is shown that under the entry-wise l∞ 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 l1 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 human myocardium gene expression data in normal and failing human hearts. The results indicate the genes in the hypertrophic cardiomyopathy pathway are less connected in the failing heart tissues than normal hearts.
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