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33
Structured variable selection with sparsityinducing norms
, 904
"... We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsityinducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual ℓ1norm and the group ℓ1norm by allowing the subsets to ov ..."
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Cited by 97 (15 self)
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We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsityinducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual ℓ1norm and the group ℓ1norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problems. We first explore the relationship between the groups defining the norm and the resulting nonzero patterns, providing both forward and backward algorithms to go back and forth from groups to patterns. This allows the design of norms adapted to specific prior knowledge expressed in terms of nonzero patterns. We also present an efficient active set algorithm, and analyze the consistency of variable selection for leastsquares linear regression in low and highdimensional settings.
Sparse Permutation Invariant Covariance Estimation
 Electronic Journal of Statistics
, 2008
"... The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in highdimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lassotype penalty. We establish a rate of convergence in the Fro ..."
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Cited by 83 (5 self)
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The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in highdimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lassotype penalty. We establish a rate of convergence in the Frobenius norm as both data dimension p and sample size n are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlationbased version of the method exhibits better rates in the operator norm. The estimator is required to be positive definite, but we avoid having to use semidefinite programming by reparameterizing the objective function
Covariance regularization by thresholding
, 2007
"... This paper considers regularizing a covariance matrix of p variables estimated from n observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or subGa ..."
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Cited by 69 (9 self)
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This paper considers regularizing a covariance matrix of p variables estimated from n observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or subGaussian, and (log p)/n → 0, and obtain explicit rates. The results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity. We discuss an intuitive resampling scheme for threshold selection and prove a general crossvalidation result that justifies this approach. We also compare thresholding to other covariance estimators in simulations and on an example from climate data. 1. Introduction. Estimation
Sparsistency and rates of convergence in large covariance matrices estimation
, 2007
"... This paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions. Here, sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probabi ..."
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Cited by 43 (5 self)
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This paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions. Here, sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one. Depending on the case of applications, sparsity priori may occur on the covariance matrix, its inverse or its Cholesky decomposition. We study these three sparsity exploration problems under a unified framework with a general penalty function. We show that the rates of convergence for these problems under the Frobenius norm are of order (sn log pn/n) 1/2, where sn is the number of nonzero elements, pn is the size of the covariance matrix and n is the sample size. This explicitly spells out the contribution of highdimensionality is merely of a logarithmic factor. The conditions on the rate with which the tuning parameter λn goes to 0 have been made explicit and compared under different penalties. As a result, for the L1penalty, to guarantee the sparsistency and optimal rate of convergence, the number of nonzero elements should be small: s ′ n = O(pn) at most, among O(p2 n) parameters, for estimating sparse covariance or correlation matrix, sparse precision or inverse correlation matrix or sparse Cholesky factor, where s ′ n is the number of the nonzero elements on the offdiagonal entries. On the other hand, using the SCAD or hardthresholding penalty functions, there is no such a restriction. 1. Introduction. Covariance
Partial Correlation Estimation by Joint Sparse Regression Models
 JASA
, 2008
"... In this article, we propose a computationally efficient approach—space (Sparse PArtial Correlation Estimation)—for selecting nonzero partial correlations under the highdimensionlowsamplesize setting. This method assumes the overall sparsity of the partial correlation matrix and employs sparse re ..."
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Cited by 37 (5 self)
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In this article, we propose a computationally efficient approach—space (Sparse PArtial Correlation Estimation)—for selecting nonzero partial correlations under the highdimensionlowsamplesize setting. This method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting. We illustrate the performance of space by extensive simulation studies. It is shown that space performs well in both nonzero partial correlation selection and the identification of hub variables, and also outperforms two existing methods. We then apply space to a microarray breast cancer dataset and identify a set of hub genes that may provide important insights on genetic regulatory networks. Finally, we prove that, under a set of suitable assumptions, the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation.
Generalized thresholding of large covariance matrices
 J. Amer. Statist. Assoc. (Theory and Methods
, 2009
"... We propose a new class of generalized thresholding operators which combine thresholding with shrinkage, and study generalized thresholding of the sample covariance matrix in high dimensions. Generalized thresholding of the covariance matrix has good theoretical properties and carries almost no compu ..."
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Cited by 21 (2 self)
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We propose a new class of generalized thresholding operators which combine thresholding with shrinkage, and study generalized thresholding of the sample covariance matrix in high dimensions. Generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden. We obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model, dimension, and the sample size, and show that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p/n → 0. In addition, we show
Network exploration via the adaptive LASSO and SCAD penalties
 Ann. Appl. Stat
, 2009
"... Graphical models are frequently used to explore networks, such as genetic networks, among a set of variables. This is usually carried out via exploring the sparsity of the precision matrix of the variables under consideration. Penalized likelihood methods are often used in such explorations. Yet, po ..."
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Cited by 18 (1 self)
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Graphical models are frequently used to explore networks, such as genetic networks, among a set of variables. This is usually carried out via exploring the sparsity of the precision matrix of the variables under consideration. Penalized likelihood methods are often used in such explorations. Yet, positivedefiniteness constraints of precision matrices make the optimization problem challenging. We introduce nonconcave penalties and the adaptive LASSO penalty to attenuate the bias problem in the network estimation. Through the local linear approximation to the nonconcave penalty functions, the problem of precision matrix estimation is recast as a sequence of penalized likelihood problems with a weighted L1 penalty and solved using the efficient algorithm of Friedman et al. [Biostatistics 9 (2008) 432–441]. Our estimation schemes are applied to two real datasets. Simulation experiments and asymptotic theory are used to justify our proposed methods. 1. Introduction. Network
Sparse inverse covariance matrix estimation via linear programming
, 2010
"... This paper considers the problem of estimating a high dimensional inverse covariance matrix that can be well approximated by “sparse ” matrices. Taking advantage of the connection between multivariate linear regression and entries of the inverse covariance matrix, we propose an estimating procedure ..."
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Cited by 12 (1 self)
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This paper considers the problem of estimating a high dimensional inverse covariance matrix that can be well approximated by “sparse ” matrices. Taking advantage of the connection between multivariate linear regression and entries of the inverse covariance matrix, we propose an estimating procedure that can effectively exploit such “sparsity”. The proposed method can be computed using linear programming and therefore has the potential to be used in very high dimensional problems. Oracle inequalities are established for the estimation error in terms of several operator norms, showing that the method is adaptive to different types of sparsity of the problem.
MultiTask Learning of Gaussian Graphical Models
"... We present multitask structure learning for Gaussian graphical models. We discuss uniqueness and boundedness of the optimal solution of the maximization problem. A block coordinate descent method leads to a provably convergent algorithm that generates a sequence of positive definite solutions. Thus ..."
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Cited by 8 (0 self)
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We present multitask structure learning for Gaussian graphical models. We discuss uniqueness and boundedness of the optimal solution of the maximization problem. A block coordinate descent method leads to a provably convergent algorithm that generates a sequence of positive definite solutions. Thus, we reduce the original problem into a sequence of strictly convex ℓ∞ regularized quadratic minimization subproblems. We further show that this subproblem leads to the continuous quadratic knapsack problem, for which very efficient methods exist. Finally, we show promising results in a dataset that captures brain function of cocaine addicted and control subjects under conditions of monetary reward. 1.
Discovering Sparse Covariance Structures with the Isomap
 Journal of Computational and Graphical Statistics
"... Regularization of covariance matrices in high dimensions is usually either based on a known ordering of variables or ignores the ordering entirely. This paper proposes a method for discovering meaningful orderings of variables based on their correlations using the Isomap, a nonlinear dimension redu ..."
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Cited by 7 (1 self)
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Regularization of covariance matrices in high dimensions is usually either based on a known ordering of variables or ignores the ordering entirely. This paper proposes a method for discovering meaningful orderings of variables based on their correlations using the Isomap, a nonlinear dimension reduction technique designed for manifold embeddings. These orderings are then used to construct a sparse covariance estimator, which is blockdiagonal and/or banded. Finding an ordering to which banding can be applied is desirable because banded estimators have been shown to be consistent in high dimensions. We show that in situations where the variables do have such a structure, the Isomap does very well at discovering it, and the resulting regularized estimator performs better for covariance estimation than other regularization methods that ignore variable order, such as thresholding. We also propose a bootstrap approach to constructing the neighborhood graph used by the Isomap, and show it leads to better estimation. We illustrate our method on data on protein consumption, where the variables (food types) have a structure but it cannot be easily described a priori, and on a gene expression data set.