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We introduce Renet, a principled generalization of the Relaxed Lasso to the Elastic Net family of estimators. While, on the one hand, $\ell_1$-regularization is a standard tool for variable selection in high-dimensional regimes and, on the…
We develop a principled way of identifying probability distributions whose independent and identically distributed (iid) realizations are compressible, i.e., can be well-approximated as sparse. We focus on Gaussian random underdetermined…
Imposition of a lasso penalty shrinks parameter estimates toward zero and performs continuous model selection. Lasso penalized regression is capable of handling linear regression problems where the number of predictors far exceeds the…
The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry…
The estimation of a precision matrix is a crucial problem in various research fields, particularly when working with high dimensional data. In such settings, the most common approach is to use the penalized maximum likelihood. The…
This paper develops asymptotic normality results for individual coordinates of robust M-estimators with convex penalty in high-dimensions, where the dimension $p$ is at most of the same order as the sample size $n$, i.e, $p/n\le\gamma$ for…
The ordinary least squares estimate in linear regression is sensitive to the influence of errors with large variance, which reduces its robustness, especially when dealing with heavy-tailed errors or outliers frequently encountered in…
Inverse problems arise in a wide spectrum of applications in fields ranging from engineering to scientific computation. Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, such…
Noiseless compressive sensing is a two-steps setting that allows for undersampling a sparse signal and then reconstructing it without loss of information. The LASSO algorithm, based on $\lone$ regularization, provides an efficient and…
Using the $\ell_1$-norm to regularize the estimation of the parameter vector of a linear model leads to an unstable estimator when covariates are highly correlated. In this paper, we introduce a new penalty function which takes into account…
We build a unifying convex analysis framework characterizing the statistical properties of a large class of penalized estimators, both under a regular and an irregular design. Our framework interprets penalized estimators as proximal…
We study the complexity of the entire regularization path for least squares regression with 1-norm penalty, known as the Lasso. Every regression parameter in the Lasso changes linearly as a function of the regularization value. The number…
The Lasso is biased. Concave penalized least squares estimation (PLSE) takes advantage of signal strength to reduce this bias, leading to sharper error bounds in prediction, coefficient estimation and variable selection. For prediction and…
This paper investigates the partial linear model by Least Absolute Deviation (LAD) regression. We parameterize the nonparametric term using Deep Neural Networks (DNNs) and formulate a penalized LAD problem for estimation. Specifically, our…
Nowadays an increasing amount of data is available and we have to deal with models in high dimension (number of covariates much larger than the sample size). Under sparsity assumption it is reasonable to hope that we can make a good…
We study the theoretical properties of the fused lasso procedure originally proposed by \cite{tibshirani2005sparsity} in the context of a linear regression model in which the regression coefficient are totally ordered and assumed to be…
This article proposes diffusion LMS strategies for distributed estimation over adaptive networks that are able to exploit sparsity in the underlying system model. The approach relies on convex regularization, common in compressive sensing,…
Dantzig selector (DS) and LASSO problems have attracted plenty of attention in statistical learning, sparse data recovery and mathematical optimization. In this paper, we provide a theoretical analysis of the sparse recovery stability of…
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…
The least absolute shrinkage and selection operator (LASSO) is a popular technique for simultaneous estimation and model selection. There have been a lot of studies on the large sample asymptotic distributional properties of the LASSO…