We consider the problem of fitting a large-scale covariance matrix to multivariate Gaussian data in such a way that the inverse is sparse, thus providing model selection. Beginning with a dense empirical covariance matrix, we solve a maximum likelihood problem with an l1-norm penalty term added to encourage sparsity in the inverse. For models with tens of nodes, the resulting problem can be solved using standard interior-point algorithms for convex optimization, but these methods scale poorly with problem size. We present two new algorithms aimed at solving problems with a thousand nodes. The first, based on Nesterov’s first-order algorithm, yields a rigorous complexity estimate for the problem, with a much better dependence on problem size than interior-point methods. Our second algorithm uses block coordinate descent, updating row/columns of the covariance matrix sequentially. Experiments with genomic data show that our method is able to uncover biologically interpretable connections among genes. 1.

We show how to apply the efficient Bayesian changepoint detection techniques of Fearnhead in the multivariate setting. We model the joint density of vector-valued observations using undirected Gaussian graphical models, whose structure we estimate. We show how we can exactly compute the MAP segmentation, as well as how to draw perfect samples from the posterior over segmentations, simultaneously accounting for uncertainty about the number and location of changepoints, as well as uncertainty about the covariance structure. We illustrate the technique by applying it to financial data and to bee tracking data. 1.

Learning of large-scale networks of interactions from microarray data is an important and challenging problem in bioinformatics. A widely used approach is to assume that the available data constitute a random sample from a multivariate distribution belonging to a Gaussian graphical model. As a consequence, the prime objects of inference are full-order partial correlations which are partial correlations between two variables given the remaining ones. In the context of microarray data the number of variables exceed the sample size and this precludes the application of traditional structure learning procedures because a sampling version of full-order partial correlations does not exist. In this paper we consider limited-order partial correlations, these are partial correlations computed on marginal distributions of manageable size, and provide a set of rules that allow one to assess the usefulness of these quantities to derive the independence structure of the underlying Gaussian graphical model. Furthermore, we introduce a novel structure learning procedure based on a quantity, obtained from limited-order partial correlations, that we call the non-rejection rate. The applicability and usefulness of the procedure are demonstrated by both simulated and real data.

We address covariance estimation under mean-squared loss in the Gaussian setting. Specifically, we consider shrinkage methods which are suitable for high dimensional problems with small number of samples (large p small n). First, we improve on the Ledoit-Wolf (LW) method by conditioning on a sufficient statistic via the Rao-Blackwell theorem, obtaining a new estimator RBLW whose mean-squared error dominates the LW under Gaussian model. Second, to further reduce the estimation error, we propose an iterative approach which approximates the clairvoyant shrinkage estimator. Convergence of this iterative method is proved and a closed form expression for the limit is determined, which is called the OAS estimator. Both of the proposed estimators have simple expressions and are easy to compute. Although the two methods are developed from different approaches, their structure is identical up to specific constants. The RBLW estimator provably dominates the LW method; and numerical simulations demonstrate that the OAS estimator performs even better, especially when n is much less than p. Index Terms — Shrinkage, covariance estimation, Rao-Blackwell, mean-squared loss

Figure 1: Trees created by users of our prototype exploratory modeling tool. Modeling times for trees in the first row ranged between 3 and 15 minutes; trees in the second row were created in 15 minutes to an hour. Enabling ordinary people to create high-quality 3D models is a long-standing problem in computer graphics. In this work, we draw from the literature on design and human cognition to better understand the design processes of novice and casual modelers, whose goals and motivations are often distinct from those of professional artists. The result is a method for creating exploratory modeling tools, which are appropriate for casual users who may lack rigidlyspecified goals or operational knowledge of modeling techniques. Our method is based on parametric design spaces, which are often high dimensional and contain wide quality variations. Our system estimates the distribution of good models in a space by tracking the modeling activity of a distributed community of users. These estimates, in turn, drive intuitive modeling tools, creating a selfreinforcing system that becomes easier to use as more people participate. We present empirical evidence that the tools developed with our method allow rapid creation of complex, high-quality 3D models by users with no specialized modeling skills or experience. We report analyses of usage patterns garnered throughout the year-long deployment of one such tool, and demonstrate the generality of the method by applying it to several design spaces.

Covariance estimation for high dimensional vectors is a classically difficult problem in statistical analysis and machine learning. In this paper, we propose a maximum likelihood (ML) approach to covariance estimation, which employs a novel sparsity constraint. More specifically, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT). The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations. Using this framework, the covariance can be efficiently estimated using greedy minimization of the log likelihood function, and the number of Givens rotations can be efficiently computed using a cross-validation procedure. The resulting estimator is positive definite and well-conditioned even when the sample size is limited. Experiments on standard hyperspectral data sets show that the SMT covariance estimate is consistently more accurate than both traditional shrinkage estimates and recently proposed graphical lasso estimates for a variety of different classes and sample sizes. 1

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost. 1 I.

We revisit the problem of feature selection in linear discriminant analysis (LDA), i.e. when features are correlated. First, we introduce a pooled centroids formulation of the multi-class LDA predictor function, in which the relative weights of Mahalanobis-tranformed predictors are given by correlation-adjusted t scores (cat scores). Second, for feature selection we propose thresholding cat scores by controlling false non-discovery rates (FNDR). We show that contrary to previous claims this FNDR procedures performs very well and similar to “higher criticism”. Third, training of the classifier function is conducted by plugin of James-Stein shrinkage estimates of correlations and variances, using analytic procedures for choosing regularization parameters. Overall, this results in an effective and computationally inexpensive framework for high-dimensional prediction with natural feature selection. The proposed shrinkage discriminant procedures are implemented in the R package “sda ” available from the R repository CRAN.

We present a procedure for effective estimation of entropy and mutual information from smallsample data, and apply it to the problem of inferring high-dimensional gene association networks. Specifically, we develop a James-Stein-type shrinkage estimator, resulting in a procedure that is highly efficient statistically as well as computationally. Despite its simplicity, we show that it outperforms eight other entropy estimation procedures across a diverse range of sampling scenarios and data-generating models, even in cases of severe undersampling. We illustrate the approach by analyzing E. coli gene expression data and computing an entropy-based gene-association network from gene expression data. A computer program is available that implements the proposed shrinkage estimator. Keywords: entropy, shrinkage estimation, James-Stein estimator, “small n, large p ” setting, mutual information, gene association network

This paper explores how entropy and other information theoretic quantities may be used to reverseengineer genetic regulatory networks from repeated microarray data. The problem of differentiating genes that undergo direct coregulation from genes whose expression is similar because they belong to the same regulatory pathway is studied from a graphical modeling viewpoint. This leads to the criteria of conditional independence which can be evaluated by computing the conditional mutual information. The latter is completely characterized by the sum of the entropies of joint variables, underlining the need for an entropy estimator that is accurate even in low sampling conditions. We introduce a new plug-in entropy estimator obtained from shrinking maximum likelihood multinomial proportions estimates to the maximum entropy target. We derive the closely related ZIPshrink and ZINBshrink entropy estimators which enhance the shrinkage estimator by first adjusting the shrinkage target depending on the fraction of structural zeros in the multinomial model. The fraction of structural zeros is estimated using a Zero-Inflated Poisson or Zero-Inflated Negative Binomial distribution to model the histogram of bin counts. We compare these three new estimators to state of the art estimators. We show that they give acceptable