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387
Regularization and variable selection via the Elastic Net
 Journal of the Royal Statistical Society, Series B
, 2005
"... Summary. We propose the elastic net, a new regularization and variable selection method. Real world data and a simulation study show that the elastic net often outperforms the lasso, while enjoying a similar sparsity of representation. In addition, the elastic net encourages a grouping effect, where ..."
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Cited by 922 (13 self)
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Summary. We propose the elastic net, a new regularization and variable selection method. Real world data and a simulation study show that the elastic net often outperforms the lasso, while enjoying a similar sparsity of representation. In addition, the elastic net encourages a grouping effect, where strongly correlated predictors tend to be in or out of the model together.The elastic net is particularly useful when the number of predictors (p) is much bigger than the number of observations (n). By contrast, the lasso is not a very satisfactory variable selection method in the p n case. An algorithm called LARSEN is proposed for computing elastic net regularization paths efficiently, much like algorithm LARS does for the lasso.
The adaptive LASSO and its oracle properties
 Journal of the American Statistical Association
"... The lasso is a popular technique for simultaneous estimation and variable selection. Lasso variable selection has been shown to be consistent under certain conditions. In this work we derive a necessary condition for the lasso variable selection to be consistent. Consequently, there exist certain sc ..."
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Cited by 660 (10 self)
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The lasso is a popular technique for simultaneous estimation and variable selection. Lasso variable selection has been shown to be consistent under certain conditions. In this work we derive a necessary condition for the lasso variable selection to be consistent. Consequently, there exist certain scenarios where the lasso is inconsistent for variable selection. We then propose a new version of the lasso, called the adaptive lasso, where adaptive weights are used for penalizing different coefficients in the!1 penalty. We show that the adaptive lasso enjoys the oracle properties; namely, it performs as well as if the true underlying model were given in advance. Similar to the lasso, the adaptive lasso is shown to be nearminimax optimal. Furthermore, the adaptive lasso can be solved by the same efficient algorithm for solving the lasso. We also discuss the extension of the adaptive lasso in generalized linear models and show that the oracle properties still hold under mild regularity conditions. As a byproduct of our theory, the nonnegative garotte is shown to be consistent for variable selection.
Sure independence screening for ultrahigh dimensional feature space
, 2006
"... Variable selection plays an important role in high dimensional statistical modeling which nowadays appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality p, estimation accuracy and computational cost are two top concerns. In a recent paper, ..."
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Cited by 279 (27 self)
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Variable selection plays an important role in high dimensional statistical modeling which nowadays appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality p, estimation accuracy and computational cost are two top concerns. In a recent paper, Candes and Tao (2007) propose the Dantzig selector using L1 regularization and show that it achieves the ideal risk up to a logarithmic factor log p. Their innovative procedure and remarkable result are challenged when the dimensionality is ultra high as the factor log p can be large and their uniform uncertainty principle can fail. Motivated by these concerns, we introduce the concept of sure screening and propose a sure screening method based on a correlation learning, called the Sure Independence Screening (SIS), to reduce dimensionality from high to a moderate scale that is below sample size. In a fairly general asymptotic framework, the SIS is shown to have the sure screening property for even exponentially growing dimensionality. As a methodological extension, an iterative SIS (ISIS) is also proposed to enhance its finite sample performance. With dimension reduced accurately from high to below sample size, variable selection can be improved on both speed and accuracy, and can then be ac
Asymptotics for Lassotype estimators
, 2000
"... this paper, we consider the asymptotic behaviour of regression estimators that minimize the residual sum of squares plus a penalty proportional to ..."
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Cited by 254 (3 self)
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this paper, we consider the asymptotic behaviour of regression estimators that minimize the residual sum of squares plus a penalty proportional to
On the LASSO and Its Dual
 Journal of Computational and Graphical Statistics
, 1999
"... Proposed by Tibshirani (1996), the LASSO (least absolute shrinkage and selection operator) estimates a vector of regression coe#cients by minimising the residual sum of squares subject to a constraint on the l 1 norm of coe#cient vector. The LASSO estimator typically has one or more zero elements ..."
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Cited by 214 (2 self)
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Proposed by Tibshirani (1996), the LASSO (least absolute shrinkage and selection operator) estimates a vector of regression coe#cients by minimising the residual sum of squares subject to a constraint on the l 1 norm of coe#cient vector. The LASSO estimator typically has one or more zero elements and thus shares characteristics of both shrinkage estimation and variable selection. In this paper we treat the LASSO as a convex programming problem and derive its dual. Consideration of the primal and dual problems together leads to important new insights into the characteristics of the LASSO estimator and to an improved method for estimating its covariance matrix. Using these results we also develop an e#cient algorithm for computing LASSO estimates which is usable even in cases where the number of regressors exceeds the number of observations. KEY WORDS AND PHRASES. Convex Programming, Dual Problem, Partial Least Squares, Quadratic Programming, Penalised Regression, Regression, Shrinkag...
Incremental Online Learning in High Dimensions
 Neural Computation
, 2005
"... Locally weighted projection regression (LWPR) is a new algorithm for incremental nonlinear function approximation in high dimensional spaces with redundant and irrelevant input dimensions. At its core, it employs nonparametric regression with locally linear models. In order to stay computationally e ..."
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Cited by 162 (18 self)
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Locally weighted projection regression (LWPR) is a new algorithm for incremental nonlinear function approximation in high dimensional spaces with redundant and irrelevant input dimensions. At its core, it employs nonparametric regression with locally linear models. In order to stay computationally e#cient and numerically robust, each local model performs the regression analysis with a small number of univariate regressions in selected directions in input space in the spirit of partial least squares regression. We discuss when and how local learning techniques can successfully work in high dimensional spaces and review the various techniques for local dimensionality reduction before finally deriving the LWPR algorithm. The properties of LWPR are that it i) learns rapidly with second order learning methods based on incremental training, ii) uses statistically sound stochastic leaveoneout cross validation for learning without the need to memorize training data, iii) adjusts its weighting kernels based only on local information in order to minimize the danger of negative interference of incremental learning, iv) has a computational complexity that is linear in the number of inputs, and v) can deal with a large number of  possibly redundant  inputs, as shown in various empirical evaluations with up to 90 dimensional data sets. For a probabilistic interpretation, predictive variance and confidence intervals are derived. To our knowledge, LWPR is the first truly incremental spatially localized learning method that can successfully and e#ciently operate in very high dimensional spaces.
Kernel partial least squares regression in reproducing kernel hilbert space
 Journal of Machine Learning Research
, 2001
"... A family of regularized least squares regression models in a Reproducing Kernel Hilbert Space is extended by the kernel partial least squares (PLS) regression model. Similar to principal components regression (PCR), PLS is a method based on the projection of input (explanatory) variables to the late ..."
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Cited by 151 (10 self)
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A family of regularized least squares regression models in a Reproducing Kernel Hilbert Space is extended by the kernel partial least squares (PLS) regression model. Similar to principal components regression (PCR), PLS is a method based on the projection of input (explanatory) variables to the latent variables (components). However, in contrast to PCR, PLS creates the components by modeling the relationship between input and output variables while maintaining most of the information in the input variables. PLS is useful in situations where the number of explanatory variables exceeds the number of observations and/or a high level of multicollinearity among those variables is assumed. Motivated by this fact we will provide a kernel PLS algorithm for construction of nonlinear regression models in possibly highdimensional feature spaces. We give the theoretical description of the kernel PLS algorithm and we experimentally compare the algorithm with the existing kernel PCR and kernel ridge regression techniques. We will demonstrate that on the data sets employed kernel PLS achieves the same results as kernel PCR but uses significantly fewer, qualitatively different components. 1.
The composite absolute penalties family for grouped and hierarchical variable selection
 Ann. Statist
"... Extracting useful information from highdimensional data is an important focus of today’s statistical research and practice. Penalized loss function minimization has been shown to be effective for this task both theoretically and empirically. With the virtues of both regularization and sparsity, the ..."
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Cited by 144 (3 self)
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Extracting useful information from highdimensional data is an important focus of today’s statistical research and practice. Penalized loss function minimization has been shown to be effective for this task both theoretically and empirically. With the virtues of both regularization and sparsity, the L1penalized squared error minimization method Lasso has been popular in regression models and beyond. In this paper, we combine different norms including L1 to form an intelligent penalty in order to add side information to the fitting of a regression or classification model to obtain reasonable estimates. Specifically, we introduce the Composite Absolute Penalties (CAP) family, which allows given grouping and hierarchical relationships between the predictors to be expressed. CAP penalties are built by defining groups and combining the properties of norm penalties at the acrossgroup and withingroup levels. Grouped selection occurs for nonoverlapping groups. Hierarchical variable selection is reached
Frequent SubStructureBased Approaches for Classifying Chemical Compounds
 In Proceedings of ICDM’03
, 2003
"... In this paper we study the problem of classifying chemical compound datasets. We present a substructurebased classification algorithm that decouples the substructure discovery process from the classification model construction and uses frequent subgraph discovery algorithms to find all topologi ..."
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Cited by 141 (6 self)
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In this paper we study the problem of classifying chemical compound datasets. We present a substructurebased classification algorithm that decouples the substructure discovery process from the classification model construction and uses frequent subgraph discovery algorithms to find all topological and geometric substructures present in the dataset. The advantage of our approach is that during classification model construction, all relevant substructures are available allowing the classifier to intelligently select the most discriminating ones. The computational scalability is ensured by the use of highly efficient frequent subgraph discovery algorithms coupled with aggressive feature selection. Our experimental evaluation on eight different classification problems shows that our approach is computationally scalable and outperforms existing schemes by 10% to 35%, on the average.