High-dimensional covariance estimation by minimizing $\ell_1$-penalized log-determinant divergence

November, 2008
Report Number: 
Pradeep Ravikumar, Martin J. Wainwright, Garvesh Raskutti and Bin Yu

Given i.i.d. observations of a random vector $X \in \mathbb{R}^p$, we study the problem of estimating both its covariance matrix $\Sigma^*$, and its inverse covariance or concentration matrix \mbox{$\Theta^* = (\Sigma^*)^{-1}$.} We estimate $\Theta^*$ by minimizing an $\ell_1$-penalized log-determinant Bregman divergence; in the multivariate Gaussian case, this approach corresponds to $\ell_1$-penalized maximum likelihood, and the structure of $\Theta^*$ is specified by the graph of an associated Gaussian Markov random field. We analyze the performance of this estimator 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 covariance matrix $\Sigma^*$; and (b) the $\ell_\infty$ operator norm of the sub-matrix $\Gamma^*_{S S}$, where $S$ indexes the graph edges, and $\Gamma^* = (\Theta^*)^{-1} \otimes (\Theta^*)^{-1}$; and (c) a mutual incoherence or irrepresentability measure on the matrix $\Gamma^*$ and (d) the rate of decay $1/\sctail(\numobs,\scdelta)$ on the probabilities $ \{|\widehat{\Sigma}^n_{ij}- \Sigma^*_{ij}| > \delta \}$, where $\widehat{\Sigma}^n$ is the sample covariance based on $n$ samples. Our first result establishes consistency of our estimate $\widehat{\Theta}$ 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 $\degmax = o(\sqrt{\spindex})$. In our second result, we show that with probability converging to one, the estimate $\widehat{\Theta}$ correctly specifies the zero pattern of the concentration matrix $\Theta^*$. We illustrate our theoretical results via simulations for various graphs and problem parameters, showing good correspondences between the theoretical predictions and behavior in simulations.

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