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. 2016 Apr;26(2):445-464.
doi: 10.5705/ss.2014.256.

Joint Estimation of Multiple High-dimensional Precision Matrices

T Tony Cai  1 , Hongzhe Li  2 , Weidong Liu  3 , Jichun Xie  4
Affiliations

Joint Estimation of Multiple High-dimensional Precision Matrices

T Tony Cai et al. Stat Sin. 2016 Apr.

Abstract

Motivated by analysis of gene expression data measured in different tissues or disease states, we consider joint estimation of multiple precision matrices 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 implemented by a second-order cone programming. Compared to separate estimation methods, the proposed joint estimation method leads to estimators converging to the true precision matrices faster. Under certain regularity conditions, the proposed procedure leads to an exact graph structure recovery with a probability tending to 1. Simulation studies show that the proposed joint estimation methods outperform other methods in graph structure recovery. The method is illustrated through an analysis of an ovarian cancer gene expression data. The results indicate that the patients with poor prognostic subtype lack some important links among the genes in the apoptosis pathway.

Keywords: Constrained optimization; Convergence rate; Graph recovery; Precision matrices; Second-order cone programming; Sparsity.

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Figures

Figure 1
Figure 1
Receiver operator characteristic curves for graph structure recovery for the simulated Erdős and Rényi graphs (the first row), and the Watts-Strogatz graphs (the second row). The x-axis and y-axis of each panel are average false positive rate and average sensitivity across K = 3 groups. Black solid line: CLIME; black dot-dashed line: GLASSO; black long-dashed line: JEMGM; grey solid line: FGL; grey dot-dashed line: GGL; grey long-dashed line: MPE.
Figure 2
Figure 2
Estimated Gaussian Graphs by the proposed method and its competitors. The dashed edges are links unique to the precision matrix estimator of the C1 subtype, the dotted edges are unique to that of other subtypes, and the solid edges are shared by both estimators. The size of node is a linear function of its degree. Upper left panel: CLIME; upper middle panel: GLASSO; upper right panel: JEMGM; bottom left panel: FGL; bottom middle panel: GGL; bottom right panel: MPE.
See this image and copyright information in PMC

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