High-dimensional covariance estimation by minimizing 1-penalized log-determinant divergence

Given i.i.d. observations of a random vector X ∈ R, we study the problem of estimating both its covariance matrix Σ, and its inverse covariance or concentration matrix Θ = (Σ). When X is multivariate Gaussian, the non-zero structure of Θ is specified by the graph of an associated Gaussian Markov random field; and a popular estimator for such sparse Θ is the l1-regularized Gaussian MLE. This estimator is sensible even for for non-Gaussian X, since it corresponds to minimizing an l1-penalized log-determinant Bregman divergence. We analyze its performance under high-dimensional scaling, in which the number of nodes in the graph p, the number of edges s, and the maximum node degree d, are allowed to grow as a function of the sample size n. In addition to the parameters (p, s, d), our analysis identifies other key quantities that control rates: (a) the l∞-operator norm of the true covariance matrix Σ; and (b) the l∞ operator norm of the sub-matrix Γ∗SS , where S indexes the graph edges, and Γ∗ = (Θ) ⊗ (Θ); and (c) a mutual incoherence or irrepresentability measure on the matrix Γ∗ and (d) the rate of decay 1/f(n, δ) on the probabilities {|Σ̂ij − Σ∗ij | > δ}, where Σ̂ is the sample covariance based on n samples. Our first result establishes consistency of our estimate Θ̂ in the elementwise maximum-norm. This in turn allows us to derive convergence rates in Frobenius and spectral norms, with improvements upon existing results for graphs with maximum node degrees d = o( √ s). In our second result, we show that with probability converging to one, the estimate Θ̂ correctly specifies the zero pattern of the concentration matrix Θ∗. We illustrate our theoretical results via simulations for various graphs and problem parameters, showing good correspondences between the theoretical predictions and behavior in simulations. AMS 2000 subject classifications: Primary 62F12; secondary 62F30.