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586
Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers
, 2010
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Regularization paths for generalized linear models via coordinate descent
, 2009
"... We develop fast algorithms for estimation of generalized linear models with convex penalties. The models include linear regression, twoclass logistic regression, and multinomial regression problems while the penalties include ℓ1 (the lasso), ℓ2 (ridge regression) and mixtures of the two (the elastic ..."
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Cited by 724 (15 self)
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We develop fast algorithms for estimation of generalized linear models with convex penalties. The models include linear regression, twoclass logistic regression, and multinomial regression problems while the penalties include ℓ1 (the lasso), ℓ2 (ridge regression) and mixtures of the two (the elastic net). The algorithms use cyclical coordinate descent, computed along a regularization path. The methods can handle large problems and can also deal efficiently with sparse features. In comparative timings we find that the new algorithms are considerably faster than competing methods.
Model Selection Through Sparse Maximum Likelihood Estimation for Multivariate Gaussian or Binary Data
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2008
"... We consider the problem of estimating the parameters of a Gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse. Our approach is to solve a maximum likelihood problem with an added ℓ1norm penalty term. The problem as formulated is convex but the memor ..."
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Cited by 334 (2 self)
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We consider the problem of estimating the parameters of a Gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse. Our approach is to solve a maximum likelihood problem with an added ℓ1norm penalty term. The problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes. We present two new algorithms for solving problems with at least a thousand nodes in the Gaussian case. Our first algorithm uses block coordinate descent, and can be interpreted as recursive ℓ1norm penalized regression. Our second algorithm, based on Nesterov’s first order method, yields a complexity estimate with a better dependence on problem size than existing interior point methods. Using a log determinant relaxation of the log partition function (Wainwright and Jordan, 2006), we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case. We test our algorithms on synthetic data, as well as on gene expression and senate voting records data.
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 164 (8 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 148 (11 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
Inferring Networks of Diffusion and Influence
, 2010
"... Information diffusion and virus propagation are fundamental processes talking place in networks. While it is often possible to directly observe when nodes become infected, observing individual transmissions (i.e., who infects whom or who influences whom) is typically very difficult. Furthermore, in ..."
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Cited by 116 (13 self)
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Information diffusion and virus propagation are fundamental processes talking place in networks. While it is often possible to directly observe when nodes become infected, observing individual transmissions (i.e., who infects whom or who influences whom) is typically very difficult. Furthermore, in many applications, the underlying network over which the diffusions and propagations spread is actually unobserved. We tackle these challenges by developing a method for tracing paths of diffusion and influence through networks and inferring the networks over which contagions propagate. Given the times when nodes adopt pieces of information or become infected, we identify the optimal network that best explains the observed infection times. Since the optimization problem is NPhard to solve exactly, we develop an efficient approximation algorithm that scales to large datasets and in practice gives provably nearoptimal performance. We demonstrate the effectiveness of our approach by tracing information cascades in a set of 170 million blogs and news articles over a one year period to infer how information flows through the online media space. We find that the diffusion network of news tends to have a coreperiphery structure with a small set of core media sites that diffuse information to the rest of the Web. These sites tend to have stable circles of influence with more general news media sites acting as connectors between them.
Sparsistency and rates of convergence in large covariance matrices estimation
, 2009
"... 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 110 (12 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.
Estimation of (near) lowrank matrices with noise and highdimensional scaling
"... We study an instance of highdimensional statistical inference in which the goal is to use N noisy observations to estimate a matrix Θ ∗ ∈ R k×p that is assumed to be either exactly low rank, or “near ” lowrank, meaning that it can be wellapproximated by a matrix with low rank. We consider an Me ..."
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Cited by 95 (14 self)
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We study an instance of highdimensional statistical inference in which the goal is to use N noisy observations to estimate a matrix Θ ∗ ∈ R k×p that is assumed to be either exactly low rank, or “near ” lowrank, meaning that it can be wellapproximated by a matrix with low rank. We consider an Mestimator based on regularization by the traceornuclearnormovermatrices, andanalyze its performance under highdimensional scaling. We provide nonasymptotic bounds on the Frobenius norm error that hold for a generalclassofnoisyobservationmodels,and apply to both exactly lowrank and approximately lowrank matrices. We then illustrate their consequences for a number of specific learning models, including lowrank multivariate or multitask regression, system identification in vector autoregressive processes, and recovery of lowrank matrices from random projections. Simulations show excellent agreement with the highdimensional scaling of the error predicted by our theory. 1.