High-dimensional Nonparanormal Graph Estimation via Smooth-projected Neighborhood Pursuit

We propose a new smooth-projected neighborhood pursuit method for estimating high dimensional undirected graphs. Our method can be viewed as a semiparametric extension of the popular neighborhood pursuit approach proposed by N. Meinshausen and P. B{ü}hlmann 2006 from Gaussian to Gaussian copula models (or the nonparanormal models as proposed by Liu et. al 2009). In terms of methodology and computation, we project a possibly indefinite symmetric matrix into the cone of positive semidefinite matrices. The projection is formulated as a smoothed element-wise $\ell_\infty$-norm minimization problem. We develop an efficient fast proximal gradient algorithm with a provable optimal rate of convergence $\cO(1/\sqrt{\epsilon})$, where $\epsilon$ is the desired accuracy for the objective value. In terms of theory, we provide an alternative view to analyze the trade-off between computational efficiency and statistical error. We give a sufficient condition to secure that the smooth-projected neighborhood pursuit estimator achieves graph estimation consistency. Empirically, we conduct real data experiments on stock and genomic datasets to illustrate the usefulness of the proposed method.