We propose Robust Lasso-Zero, an extension of the Lasso-Zero methodology, initially introduced for sparse linear models, to the sparse corruptions problem. We give theoretical guarantees on the sign recovery of the parameters for a slightly simplified version of the estimator, called Thresholded Justice Pursuit. The use of Robust Lasso-Zero is showcased for variable selection with missing values in the covariates. In addition to not requiring the specification of a model for the covariates, nor estimating their covariance matrix or the noise variance, the method has the great advantage of handling missing not-at random values without specifying a parametric model. Numerical experiments and a medical application underline the relevance of Robust Lasso-Zero in such a context with few available competitors. The method is easy to use and implemented in the R library lass0.
Given data $\mathbf{y}$ and $k$ covariates $\mathbf{x}_j$ one problem in linear regression is to decide which if any of the covariates to include when regressing the dependent variable $\mathbf{y}$ on the covariates $\mathbf{x}_j$. In this paper three such methods, lasso, knockoff and Gaussian covariates are compared using simulations and real data. The Gaussian covariate method is based on exact probabilities which are valid for all $\mathbf{y}$ and $\mathbf{x}_j$ making it model free. Moreover the probabilities agree with those based on the F-distribution for the standard linear model with i…
Lasso is a popular and efficient approach to simultaneous estimation and variable selection in high-dimensional regression models. In this paper, a robust LAD-lasso method for multiple outcomes is presented that addresses the challenges of non-normal outcome distributions and outlying observations. Measured covariate data from space or time, or spectral bands or genomic positions often have natural correlation structure arising from measuring distance between the covariates. The proposed multi-outcome approach includes handling of such covariate blocks by a group fusion penalty, which encourag…
We study the limitations of the well known LASSO regression as a variable selector when there exists dependence structures among covariates. We analyze both the classic situation with $n\geq p$ and the high dimensional framework with $p>n$. Restrictive properties of this methodology to guarantee optimality, as well as the inconveniences in practice, are analyzed. Examples of these drawbacks are showed by means of a extensive simulation study, making use of different dependence scenarios. In order to search for improvements, a broad comparison with LASSO derivatives and alternatives is carried …
Two extensions of generalized linear models are considered. In the first one, response variables depend on multiple linear combinations of covariates. In the second one, only response variables are observed while the linear covariates are missing. We derive stochastic Lipschitz continuity results for the loss functions involved in the regression problems and apply them to get bounds on estimation error for Lasso. Multivariate comparison results on Rademacher complexity are obtained as tools to establish the stochastic Lipschitz continuity results.
Regression with the lasso penalty is a popular tool for performing dimension reduction when the number of covariates is large. In many applications of the lasso, like in genomics, covariates are subject to measurement error. We study the impact of measurement error on linear regression with the lasso penalty, both analytically and in simulation experiments. A simple method of correction for measurement error in the lasso is then considered. In the large sample limit, the corrected lasso yields sign consistent covariate selection under conditions very similar to the lasso with perfect measureme…
We propose a new measure of variable importance in high-dimensional regression based on the change in the LASSO solution path when one covariate is left out. The proposed procedure provides a novel way to calculate variable importance and conduct variable screening. In addition, our procedure allows for the construction of P-values for testing whether each coefficient is equal to zero as well as for testing hypotheses involving multiple regression coefficients simultaneously; bootstrap techniques are used to construct the null distribution. For low-dimensional linear models, our method can ach…
Although the Lasso has been extensively studied, the relationship between its prediction performance and the correlations of the covariates is not fully understood. In this paper, we give new insights into this relationship in the context of multiple linear regression. We show, in particular, that the incorporation of a simple correlation measure into the tuning parameter can lead to a nearly optimal prediction performance of the Lasso even for highly correlated covariates. However, we also reveal that for moderately correlated covariates, the prediction performance of the Lasso can be mediocr…
Stabilizing Private LASSO under Heterogeneous Covariates via Anisotropic Objective Perturbation
Department of Information Science Ochanomizu University Bunkyo-ku, Tokyo, Japan（信息科学系, 陈仓区, 东京, 日本, 东方大学）；RIKEN center for AIP Chuo-ku, Tokyo, Japan（理化学研究所AIP研究中心, 东京都, 东京, 日本）
We propose a computationally intensive method, the random lasso method, for variable selection in linear models. The method consists of two major steps. In step 1, the lasso method is applied to many bootstrap samples, each using a set of randomly selected covariates. A measure of importance is yielded from this step for each covariate. In step 2, a similar procedure to the first step is implemented with the exception that for each bootstrap sample, a subset of covariates is randomly selected with unequal selection probabilities determined by the covariates' importance. Adaptive lasso may be u…
Least Absolute Shrinkage and Selection Operator or the Lasso, introduced by Tibshirani (1996), is a popular estimation procedure in multiple linear regression when underlying design has a sparse structure, because of its property that it sets some regression coefficients exactly equal to 0. In this article, we develop a perturbation bootstrap method and establish its validity in approximating the distribution of the Lasso in heteroscedastic linear regression. We allow the underlying covariates to be either random or non-random. We show that the proposed bootstrap method works irrespective of t…
In the high-dimensional regression model a response variable is linearly related to $p$ covariates, but the sample size $n$ is smaller than $p$. We assume that only a small subset of covariates is `active' (i.e., the corresponding coefficients are non-zero), and consider the model-selection problem of identifying the active covariates. A popular approach is to estimate the regression coefficients through the Lasso ($\ell_1$-regularized least squares). This is known to correctly identify the active set only if the irrelevant covariates are roughly orthogonal to the relevant ones, as quantified …
A comparative study of augmented inverse propensity weighted estimators using outcome-oriented covariate selection via penalization with outcome-adaptive lasso
The purpose of this article is to provide an adaptive estimator of the baseline function in the Cox model with high-dimensional covariates. We consider a two-step procedure : first, we estimate the regression parameter of the Cox model via a Lasso procedure based on the partial log-likelihood, secondly, we plug this Lasso estimator into a least-squares type criterion and then perform a model selection procedure to obtain an adaptive penalized contrast estimator of the baseline function. Using non-asymptotic estimation results stated for the Lasso estimator of the regression parameter, we estab…
Recent work has focused on the problem of conducting linear regression when the number of covariates is very large, potentially greater than the sample size. To facilitate this, one useful tool is to assume that the model can be well approximated by a fit involving only a small number of covariates -- a so called sparsity assumption, which leads to the Lasso and other methods. In many situations, however, the covariates can be considered to be structured, in that the selection of some variables favours the selection of others -- with variables organised into groups entering or leaving the mode…
We propose a new approach to safe variable preselection in high-dimensional penalized regression, such as the lasso. Preselection - to start with a manageable set of covariates - has often been implemented without clear appreciation of its potential bias. Based on sequential implementation of the lasso with increasing lists of predictors, we find a new property of the set of corresponding cross-validation curves, a pattern that we call freezing. It allows to determine a subset of covariates with which we reach the same lasso solution as would be obtained using the full set of covariates. Freez…
A network lasso enables us to construct a model for each sample, which is known as multi-task learning. Existing methods for multi-task learning cannot be applied to compositional data due to their intrinsic properties. In this paper, we propose a multi-task learning method for compositional data using a sparse network lasso. We focus on a symmetric form of the log-contrast model, which is a regression model with compositional covariates. The effectiveness of the proposed method is shown through simulation studies and application to gut microbiome data.
We consider the estimation of regression models on strata defined using a categorical covariate, in order to identify interactions between this categorical covariate and the other predictors. A basic approach requires the choice of a reference stratum. We show that the performance of a penalized version of this approach depends on this arbitrary choice. We propose a refined approach that bypasses this arbitrary choice, at almost no additional computational cost. Regarding model selection consistency, our proposal mimics the strategy based on an optimal and covariate-specific choice for the ref…
We consider the problem of estimating a low-dimensional parameter in high-dimensional linear regression. Constructing an approximately unbiased estimate of the parameter of interest is a crucial step towards performing statistical inference. Several authors suggest to orthogonalize both the variable of interest and the outcome with respect to the nuisance variables, and then regress the residual outcome with respect to the residual variable. This is possible if the covariance structure of the regressors is perfectly known, or is sufficiently structured that it can be estimated accurately from …
Adaptive LASSO Penalized Minimum Density Power Divergence Estimation through Least Squares Approximation: Application to Bone Mineral Density Data from the SWAN Study
Indian Institute of Technology (Indian School of Mines)（印度理工学院（印度矿业学院））
We consider the linear regression problem. We propose the S-Lasso procedure to estimate the unknown regression parameters. This estimator enjoys sparsity of the representation while taking into account correlation between successive covariates (or predictors). The study covers the case when $p\gg n$, i.e. the number of covariates is much larger than the number of observations. In the theoretical point of view, for fixed $p$, we establish asymptotic normality and consistency in variable selection results for our procedure. When $p\geq n$, we provide variable selection consistency results and sh…
In this work, we consider a manufactory process which can be described by a multiple-instance logistic regression model. In order to compute the maximum likelihood estimation of the unknown coefficient, an expectation-maximization algorithm is proposed, and the proposed modeling approach can be extended to identify the important covariates by adding the coefficient penalty term into the likelihood function. In addition to essential technical details, we demonstrate the usefulness of the proposed method by simulations and real examples.
We consider the problem of predicting an outcome variable using $p$ covariates that are measured on $n$ independent observations, in the setting in which flexible and interpretable fits are desirable. We propose the fused lasso additive model (FLAM), in which each additive function is estimated to be piecewise constant with a small number of adaptively-chosen knots. FLAM is the solution to a convex optimization problem, for which a simple algorithm with guaranteed convergence to the global optimum is provided. FLAM is shown to be consistent in high dimensions, and an unbiased estimator of its …
Inferring causal relationships or related associations from observational data can be invalidated by the existence of hidden confounding. We focus on a high-dimensional linear regression setting, where the measured covariates are affected by hidden confounding and propose the {\em Doubly Debiased Lasso} estimator for individual components of the regression coefficient vector. Our advocated method simultaneously corrects both the bias due to estimation of high-dimensional parameters as well as the bias caused by the hidden confounding. We establish its asymptotic normality and also prove that i…
INRIA（法国国家信息与自动化研究所）；Laboratoire d’Informatique de l’École Normale Supérieure（巴黎高等师范学院计算机实验室）
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 the correlation of the design matrix to stabilize the estimation. This norm, called the trace Lasso, uses the trace norm, which is a convex surrogate of the rank, of the selected covariates as the criterion of model complexity. We analyze the properties of our norm, describe an optimization algorithm based on reweighted least-squares, and illustrate the beh…
University of Science and Technology of China（中国科学技术大学）；Peking University（北京大学）；Central China Normal University（华中师范大学）
When we are interested in high-dimensional system and focus on classification performance, the $\ell_{1}$-penalized logistic regression is becoming important and popular. However, the Lasso estimates could be problematic when penalties of different coefficients are all the same and not related to the data. We proposed two types of weighted Lasso estimates depending on covariates by the McDiarmid inequality. Given sample size $n$ and dimension of covariates $p$, the finite sample behavior of our proposed methods with a diverging number of predictors is illustrated by non-asymptotic oracle inequ…
We analyze the statistical consistency of robust estimators for precision matrices in high dimensions. We focus on a contamination mechanism acting cellwise on the data matrix. The estimators we analyze are formed by plugging appropriately chosen robust covariance matrix estimators into the graphical Lasso and CLIME. Such estimators were recently proposed in the robust statistics literature, but only analyzed mathematically from the point of view of the breakdown point. This paper provides complementary high-dimensional error bounds for the precision matrix estimators that reveal the interplay…
We introduce a novel regression model for the conditional left and right tail of a possibly heavy-tailed response. The proposed model can be used to learn the effect of covariates on an extreme value setting via a Lasso-type specification based on a Lagrangian restriction. Our model can be used to track if some covariates are significant for the lower values, but not for the (right) tail---and vice-versa; in addition to this, the proposed model bypasses the need for conditional threshold selection in an extreme value theory framework. We assess the finite-sample performance of the proposed met…
Gaussian graphical regression is a powerful means that regresses the precision matrix of a Gaussian graphical model on covariates, permitting the numbers of the response variables and covariates to far exceed the sample size. Model fitting is typically carried out via separate node-wise lasso regressions, ignoring the network-induced structure among these regressions. Consequently, the error rate is high, especially when the number of nodes is large. We propose a multi-task learning estimator for fitting Gaussian graphical regression models; we design a cross-task group sparsity penalty and a …
This paper studies the problem of accurately recovering a sparse vector $β^{\star}$ from highly corrupted linear measurements $y = X β^{\star} + e^{\star} + w$ where $e^{\star}$ is a sparse error vector whose nonzero entries may be unbounded and $w$ is a bounded noise. We propose a so-called extended Lasso optimization which takes into consideration sparse prior information of both $β^{\star}$ and $e^{\star}$. Our first result shows that the extended Lasso can faithfully recover both the regression as well as the corruption vector. Our analysis relies on the notion of extended restricted eigen…
Logistic lasso regression with nearest neighbors for gradient-based dimension reduction
University of Oslo（奥斯陆大学）；CREST, ENSAI, University of Rennes（法国国家经济统计与分析中心，雷恩高等应用统计学院，雷恩大学）；Univ Angers, CNRS, LAREMA, SFR MATHSTIC（昂热大学，法国国家科学研究中心，拉雷马实验室，数学科学联合研究服务）
We consider the problems of variable selection and estimation in nonparametric additive regression models for high-dimensional data. In recent years, several methods have been proposed to model nonlinear relationships when the number of covariates exceeds the number of observations by using spline basis functions and group penalties. Nonlinear {\it monotone} effects on the response play a central role in many situations, in particular in medicine and biology. We construct the monotone splines lasso (MS-lasso) to select variables and estimate effects using monotone spline (I-splines). The addit…
Effect modification occurs when the effect of the treatment on an outcome varies according to the level of other covariates and often has important implications in decision making. When there are tens or hundreds of covariates, it becomes necessary to use the observed data to select a simpler model for effect modification and then make valid statistical inference. We propose a two stage procedure to solve this problem. First, we use Robinson's transformation to decouple the nuisance parameters from the treatment effect of interest and use machine learning algorithms to estimate the nuisance pa…
This paper considers errors-in-variables models in a high-dimensional setting where the number of covariates can be much larger than the sample size, and there are only a small number of non-zero covariates. The presence of measurement error in the covariates can result in severely biased parameter estimates, and also affects the ability of penalized methods such as the lasso to recover the true sparsity pattern. A new estimation procedure called SIMSELEX (SIMulation-SELection-EXtrapolation) is proposed. This procedure augments the traditional SIMEX approach with a variable selection step base…
Modern soil mapping is characterised by the need to interpolate samples of geostatistical response observations and the availability of relatively large numbers of environmental characteristics for consideration as covariates to aid this interpolation. We demonstrate the efficiency of the Least Angle Regression algorithm for Least Absolute Shrinkage and Selection Operator (LASSO) penalized multiple linear regression at selecting covariates to aid the spatial interpolation of geostatistical soil carbon observations under an ultrahigh dimensional scenario. Where an exhaustive search of the model…
Administrative data, or non-probability sample data, are increasingly being used to obtain official statistics due to their many benefits over survey methods. In particular, they are less costly, provide a larger sample size, and are not reliant on the response rate. However, it is difficult to obtain an unbiased estimate of the population mean from such data due to the absence of design weights. Several estimation approaches have been proposed recently using an auxiliary probability sample which provides representative covariate information of the target population. However, when this covaria…
Often in public health, we are interested in the treatment effect of an intervention on a population that is systemically different from the experimental population the intervention was originally evaluated in. When treatment effect heterogeneity is present in a randomized controlled trial, generalizing the treatment effect from this experimental population to a target population of interest is a complex problem; it requires the characterization of both the treatment effect heterogeneity and the baseline covariate mismatch between the two populations. Despite the importance of this problem, th…
Lasso and Dantzig selector are standard procedures able to perform variable selection and estimation simultaneously. This paper is concerned with extending these procedures to spatial point process intensity estimation. We propose adaptive versions of these procedures, develop efficient computational methodologies and derive asymptotic results for a large class of spatial point processes under an original setting where the number of parameters, i.e. the number of spatial covariates considered, increases with the expected number of data points. Both procedures are compared theoretically, in a s…
Selection of important covariates and to drop the unimportant ones from a high-dimensional regression model is a long standing problem and hence have received lots of attention in the last two decades. After selecting the correct model, it is also important to properly estimate the existing parameters corresponding to important covariates. In this spirit, Fan and Li (2001) proposed Oracle property as a desired feature of a variable selection method. Oracle property has two parts; one is the variable selection consistency (VSC) and the other one is the asymptotic normality. Keeping VSC fixed an…
The Lasso is a popular statistical tool invented by Robert Tibshirani for linear regression when the number of covariates is greater than or comparable to the number of observations. The purpose of this note is to highlight the simple fact (noted in a number of earlier papers in various guises) that for the loss function considered in Tibshirani's original paper, the Lasso is consistent under almost no assumptions at all.
We consider a high-dimensional regression model with a possible change-point due to a covariate threshold and develop the Lasso estimator of regression coefficients as well as the threshold parameter. Our Lasso estimator not only selects covariates but also selects a model between linear and threshold regression models. Under a sparsity assumption, we derive non-asymptotic oracle inequalities for both the prediction risk and the $\ell_1$ estimation loss for regression coefficients. Since the Lasso estimator selects variables simultaneously, we show that oracle inequalities can be established w…
We present a novel method for variable selection in regression models when covariates are measured with error. The iterative algorithm we propose, MEBoost, follows a path defined by estimating equations that correct for covariate measurement error. Via simulation, we evaluated our method and compare its performance to the recently-proposed Convex Conditioned Lasso (CoCoLasso) and to the "naive" Lasso which does not correct for measurement error. Increasing the degree of measurement error increased prediction error and decreased the probability of accurate covariate selection, but this loss of …
The instability in the selection of models is a major concern with data sets containing a large number of covariates. We focus on stability selection which is used as a technique to improve variable selection performance for a range of selection methods, based on aggregating the results of applying a selection procedure to sub-samples of the data where the observations are subject to right censoring. The accelerated failure time (AFT) models have proved useful in many contexts including the heavy censoring (as for example in cancer survival) and the high dimensionality (as for example in micro…
Harvard University（哈佛大学）；University of California at Berkeley（加州大学伯克利分校）；University of California at San Francisco（加州大学旧金山分校）；Preva Group（Preva集团）
We propose a novel, fully nonparametric approach for the multi-task learning, the Multi-task Highly Adaptive Lasso (MT-HAL). MT-HAL simultaneously learns features, samples and task associations important for the common model, while imposing a shared sparse structure among similar tasks. Given multiple tasks, our approach automatically finds a sparse sharing structure. The proposed MTL algorithm attains a powerful dimension-free convergence rate of $o_p(n^{-1/4})$ or better. We show that MT-HAL outperforms sparsity-based MTL competitors across a wide range of simulation studies, including setti…
We introduce a pliable lasso method for estimation of interaction effects in the Cox proportional hazards model framework. The pliable lasso is a linear model that includes interactions between covariates X and a set of modifying variables Z and assumes sparsity of the main effects and interaction effects. The hierarchical penalty excludes interaction effects when the corresponding main effects are zero: this avoids overfitting and an explosion of model complexity. We extend this method to the Cox model for survival data, incorporating modifiers that are either fixed or varying in time into th…
Statistical inference after variable selection in Cox models: A simulation study
Mathematical Statistics and Artificial Intelligence in Medicine, University of Augsburg（医学中的数学统计与人工智能，乌尔姆大学）；Center for Advanced Analytics and Predictive Sciences (CAAPS), University of Augsburg（高级分析与预测科学中心（CAAPS），乌尔姆大学）
We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true regression coefficients. Similar results based on an extension of the restricted eigenvalue can be also proved by our method. However, the presented oracle inequalities are sharper since the compatibility and cone invertibility fac…
