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Two important goals of high-dimensional modeling are prediction and variable selection. In this article, we consider regularization with combined $L_1$ and concave penalties, and study the sampling properties of the global optimum of the…
High-dimensional sparse modeling via regularization provides a powerful tool for analyzing large-scale data sets and obtaining meaningful, interpretable models. The use of nonconvex penalty functions shows advantage in selecting important…
Popular regularizers with non-differentiable penalties, such as Lasso, Elastic Net, Generalized Lasso, or SLOPE, reduce the dimension of the parameter space by inducing sparsity or clustering in the estimators' coordinates. In this paper,…
High-dimensional sparse modeling with censored survival data is of great practical importance, as exemplified by modern applications in high-throughput genomic data analysis and credit risk analysis. In this article, we propose a class of…
Penalized likelihood methods are fundamental to ultra-high dimensional variable selection. How high dimensionality such methods can handle remains largely unknown. In this paper, we show that in the context of generalized linear models,…
Regularization is widely used in statistics and machine learning to prevent overfitting and gear solution towards prior information. In general, a regularized estimation problem minimizes the sum of a loss function and a penalty term. The…
Penalized least squares estimation is a popular technique in high-dimensional statistics. It includes such methods as the LASSO, the group LASSO, and the nuclear norm penalized least squares. The existing theory of these methods is not…
High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding need for better measurement specific model selection. In this paper we…
This paper develops a general theory on rates of convergence of penalized spline estimators for function estimation when the likelihood functional is concave in candidate functions, where the likelihood is interpreted in a broad sense that…
A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by Fan and Li to simultaneously estimate parameters and select important variables. They demonstrated that this class of…
The Lasso has attracted the attention of many authors these last years. While many efforts have been made to prove that the Lasso behaves like a variable selection procedure at the price of strong (though unavoidable) assumptions on the…
Model selection and sparse recovery are two important problems for which many regularization methods have been proposed. We study the properties of regularization methods in both problems under the unified framework of regularized least…
The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso, square-root lasso, elastic net, and many others. In this paper, we establish a general…
Learning sparse models from data is an important task in all those frameworks where relevant information should be identified within a large dataset. This can be achieved by formulating and solving suitable sparsity promoting optimization…
The $\ell_1$-penalized method, or the Lasso, has emerged as an important tool for the analysis of large data sets. Many important results have been obtained for the Lasso in linear regression which have led to a deeper understanding of…
This paper compares convex and non-convex penalized likelihood methods in high-dimensional statistical modeling, focusing on their strengths and limitations. Convex penalties, like LASSO, offer computational efficiency and strong…
We consider high-dimensional generalized linear models with Lipschitz loss functions, and prove a nonasymptotic oracle inequality for the empirical risk minimizer with Lasso penalty. The penalty is based on the coefficients in the linear…
In sparse linear regression, the SLOPE estimator generalizes LASSO by penalizing different coordinates of the estimate according to their magnitudes. In this paper, we present a precise performance characterization of SLOPE in the…
We provide theoretical analysis of the statistical and computational properties of penalized $M$-estimators that can be formulated as the solution to a possibly nonconvex optimization problem. Many important estimators fall in this…
This paper proposes a general framework for penalized convex empirical criteria and a new version of the Sparse-Group LASSO (SGL, Simon and al., 2013), called the adaptive SGL, where both penalties of the SGL are weighted by preliminary…