High-dimensional Sparse Inverse Covariance Estimation using Greedy Methods

In this paper we consider the task of estimating the non-zero pattern of the sparse inverse covariance matrix of a zero-mean Gaussian random vector from a set of iid samples. Note that this is also equivalent to recovering the underlying graph structure of a sparse Gaussian Markov Random Field (GMRF). We present two novel greedy approaches to solving this problem. The first estimates the non-zero covariates of the overall inverse covariance matrix using a series of global forward and backward greedy steps. The second estimates the neighborhood of each node in the graph separately, again using greedy forward and backward steps, and combines the intermediate neighborhoods to form an overall estimate. The principal contribution of this paper is a rigorous analysis of the sparsistency, or consistency in recovering the sparsity pattern of the inverse covariance matrix. Surprisingly, we show that both the local and global greedy methods learn the full structure of the model with high probability given just $O(d\log(p))$ samples, which is a \emph{significant} improvement over state of the art $\ell_1$-regularized Gaussian MLE (Graphical Lasso) that requires $O(d^2\log(p))$ samples. Moreover, the restricted eigenvalue and smoothness conditions imposed by our greedy methods are much weaker than the strong irrepresentable conditions required by the $\ell_1$-regularization based methods. We corroborate our results with extensive simulations and examples, comparing our local and global greedy methods to the $\ell_1$-regularized Gaussian MLE as well as the Neighborhood Greedy method to that of nodewise $\ell_1$-regularized linear regression (Neighborhood Lasso).Comment: Accepted to AI STAT 2012 for Oral Presentatio

Regression modeling on stratified data with the lasso

We consider the estimation of regression models on strata defined using a categorical covariate, in order to identify interactions between this categorical covariate and the other predictors. A basic approach requires the choice of a reference stratum. We show that the performance of a penalized version of this approach depends on this arbitrary choice. We propose a refined approach that bypasses this arbitrary choice, at almost no additional computational cost. Regarding model selection consistency, our proposal mimics the strategy based on an optimal and covariate-specific choice for the reference stratum. Results from an empirical study confirm that our proposal generally outperforms the basic approach in the identification and description of the interactions. An illustration on gene expression data is provided.Comment: 23 pages, 5 figure

A sparse conditional Gaussian graphical model for analysis of genetical genomics data

Genetical genomics experiments have now been routinely conducted to measure both the genetic markers and gene expression data on the same subjects. The gene expression levels are often treated as quantitative traits and are subject to standard genetic analysis in order to identify the gene expression quantitative loci (eQTL). However, the genetic architecture for many gene expressions may be complex, and poorly estimated genetic architecture may compromise the inferences of the dependency structures of the genes at the transcriptional level. In this paper we introduce a sparse conditional Gaussian graphical model for studying the conditional independent relationships among a set of gene expressions adjusting for possible genetic effects where the gene expressions are modeled with seemingly unrelated regressions. We present an efficient coordinate descent algorithm to obtain the penalized estimation of both the regression coefficients and the sparse concentration matrix. The corresponding graph can be used to determine the conditional independence among a group of genes while adjusting for shared genetic effects. Simulation experiments and asymptotic convergence rates and sparsistency are used to justify our proposed methods. By sparsistency, we mean the property that all parameters that are zero are actually estimated as zero with probability tending to one. We apply our methods to the analysis of a yeast eQTL data set and demonstrate that the conditional Gaussian graphical model leads to a more interpretable gene network than a standard Gaussian graphical model based on gene expression data alone.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS494 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Graphical Models via Univariate Exponential Family Distributions

Undirected graphical models, or Markov networks, are a popular class of statistical models, used in a wide variety of applications. Popular instances of this class include Gaussian graphical models and Ising models. In many settings, however, it might not be clear which subclass of graphical models to use, particularly for non-Gaussian and non-categorical data. In this paper, we consider a general sub-class of graphical models where the node-wise conditional distributions arise from exponential families. This allows us to derive multivariate graphical model distributions from univariate exponential family distributions, such as the Poisson, negative binomial, and exponential distributions. Our key contributions include a class of M-estimators to fit these graphical model distributions; and rigorous statistical analysis showing that these M-estimators recover the true graphical model structure exactly, with high probability. We provide examples of genomic and proteomic networks learned via instances of our class of graphical models derived from Poisson and exponential distributions.Comment: Journal of Machine Learning Researc

Stability Approach to Regularization Selection (StARS) for High Dimensional Graphical Models

A challenging problem in estimating high-dimensional graphical models is to choose the regularization parameter in a data-dependent way. The standard techniques include $K$-fold cross-validation ($K$-CV), Akaike information criterion (AIC), and Bayesian information criterion (BIC). Though these methods work well for low-dimensional problems, they are not suitable in high dimensional settings. In this paper, we present StARS: a new stability-based method for choosing the regularization parameter in high dimensional inference for undirected graphs. The method has a clear interpretation: we use the least amount of regularization that simultaneously makes a graph sparse and replicable under random sampling. This interpretation requires essentially no conditions. Under mild conditions, we show that StARS is partially sparsistent in terms of graph estimation: i.e. with high probability, all the true edges will be included in the selected model even when the graph size diverges with the sample size. Empirically, the performance of StARS is compared with the state-of-the-art model selection procedures, including $K$-CV, AIC, and BIC, on both synthetic data and a real microarray dataset. StARS outperforms all these competing procedures