This paper studies inference in the high-dimensional linear regression model with outliers. Sparsity constraints are imposed on the vector of coefficients of the covariates. The number of outliers can grow with the sample size while their proportion goes to 0. We propose a two-step procedure for inference on the coefficients of a fixed subset of regressors. The first step is a based on several square-root lasso l1-norm penalized estimators, while the second step is the ordinary least squares estimator applied to a well chosen regression. We establish asymptotic normality of the two-step estimator. The proposed procedure is efficient in the sense that it attains the semiparametric efficiency bound when applied to the model without outliers under homoscedasticity. This approach is also computationally advantageous, it amounts to solving a finite number of convex optimization programs.
This paper considers the problem of inference in a linear regression model with outliers where the number of outliers can grow with sample size but their proportion goes to 0. We apply the square-root lasso estimator penalizing the l1-norm of a random vector which is non-zero for outliers. We derive rates of convergence and asymptotic normality. Our estimator has the same asymptotic variance as the OLS estimator in the standard linear model. This enables to build tests and confidence sets in the usual and simple manner. The proposed procedure is also computationally advantageous as it amounts …
We study functional regression with random subgaussian design and real-valued response. The focus is on the problems in which the regression function can be well approximated by a functional linear model with the slope function being "sparse" in the sense that it can be represented as a sum of a small number of well separated "spikes". This can be viewed as an extension of now classical sparse estimation problems to the case of infinite dictionaries. We study an estimator of the regression function based on penalized empirical risk minimization with quadratic loss and the complexity penalty de…
This paper deals with the trace regression model where $n$ entries or linear combinations of entries of an unknown $m_1\times m_2$ matrix $A_0$ corrupted by noise are observed. We propose a new nuclear norm penalized estimator of $A_0$ and establish a general sharp oracle inequality for this estimator for arbitrary values of $n,m_1,m_2$ under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form and we prove that it satisfies oracle inequalities with faster rates of convergence than in t…
Adaptive nuclear-norm penalization is proposed for low-rank matrix approximation, by which we develop a new reduced-rank estimation method for the general high-dimensional multivariate regression problems. The adaptive nuclear norm of a matrix is defined as the weighted sum of the singular values of the matrix. For example, the pre-specified weights may be some negative power of the singular values of the data matrix (or its projection in regression setting). The adaptive nuclear norm is generally non-convex under the natural restriction that the weight decreases with the singular value. Howev…
The prevalence of data collected on the same set of samples from multiple sources (i.e., multi-view data) has prompted significant development of data integration methods based on low-rank matrix factorizations. These methods decompose signal matrices from each view into the sum of shared and individual structures, which are further used for dimension reduction, exploratory analyses, and quantifying associations across views. However, existing methods have limitations in modeling partially-shared structures due to either too restrictive models, or restrictive identifiability conditions. To add…
The matrix completion problem consists in reconstructing a matrix from a sample of entries, possibly observed with noise. A popular class of estimator, known as nuclear norm penalized estimators, are based on minimizing the sum of a data fitting term and a nuclear norm penalization. Here, we investigate the case where the noise distribution belongs to the exponential family and is sub-exponential. Our framework alllows for a general sampling scheme. We first consider an estimator defined as the minimizer of the sum of a log-likelihood term and a nuclear norm penalization and prove an upper bou…
Originally developed for imputing missing entries in low rank, or approximately low rank matrices, matrix completion has proven widely effective in many problems where there is no reason to assume low-dimensional linear structure in the underlying matrix, as would be imposed by rank constraints. In this manuscript, we build some theoretical intuition for this behavior. We consider matrices which are not necessarily low-rank, but lie in a low-dimensional non-linear manifold. We show that nuclear-norm penalization is still effective for recovering these matrices when observations are missing com…
Sparse covariates are frequent in classification and regression problems and in these settings the task of variable selection is usually of interest. As it is well known, sparse statistical models correspond to situations where there are only a small number of non--zero parameters and for that reason, they are much easier to interpret than dense ones. In this paper, we focus on the logistic regression model and our aim is to address robust and penalized estimation for the regression parameter. We introduce a family of penalized weighted $M-$type estimators for the logistic regression parameter…
The problem of estimating ARMA models is computationally interesting due to the nonconcavity of the log-likelihood function. Recent results were based on the convex minimization. Joint model selection using penalization by a convex norm, e.g. the nuclear norm of a certain matrix related to the state space formulation was extensively studied from a computational viewpoint. The goal of the present short note is to present a theoretical study of a nuclear norm penalization based variant of the method of \cite{Bauer:Automatica05,Bauer:EconTh05} under the assumption of a Gaussian noise process.
This paper provides a comprehensive estimation framework via nuclear norm plus $l_1$ norm penalization for high-dimensional approximate factor models with a sparse residual covariance. The underlying assumptions allow for non-pervasive latent eigenvalues and a prominent residual covariance pattern. In that context, existing approaches based on principal components may lead to misestimate the latent rank, due to the numerical instability of sample eigenvalues. On the contrary, the proposed optimization problem retrieves the latent covariance structure and exactly recovers the latent rank and th…
Capital University of Economics and Business（首都经济贸易大学）；Hainan University（海南大学）；China Agricultural University（中国农业大学）
Classical principal component analysis (PCA) may suffer from the sensitivity to outliers and noise. Therefore PCA based on $\ell_1$-norm and $\ell_p$-norm ($0 < p < 1$) have been studied. Among them, the ones based on $\ell_p$-norm seem to be most interesting from the robustness point of view. However, their numerical performance is not satisfactory. Note that, although T$\ell_1$-norm is similar to $\ell_p$-norm ($0 < p < 1$) in some sense, it has the stronger suppression effect to outliers and better continuity. So PCA based on T$\ell_1$-norm is proposed in this paper. Our numerical experimen…
In high-throughput genetics studies, an important aim is to identify gene-environment interactions associated with the clinical outcomes. Recently, multiple marginal penalization methods have been developed and shown to be effective in G$\times$E studies. However, within the Bayesian framework, marginal variable selection has not received much attention. In this study, we propose a novel marginal Bayesian variable selection method for G$\times$E studies. In particular, our marginal Bayesian method is robust to data contamination and outliers in the outcome variables. With the incorporation of …
We study Empirical Risk Minimizers (ERM) and Regularized Empirical Risk Minimizers (RERM) for regression problems with convex and $L$-Lipschitz loss functions. We consider a setting where $|\cO|$ malicious outliers contaminate the labels. In that case, under a local Bernstein condition, we show that the $L_2$-error rate is bounded by $ r_N + AL |\cO|/N$, where $N$ is the total number of observations, $r_N$ is the $L_2$-error rate in the non-contaminated setting and $A$ is a parameter coming from the local Bernstein condition. When $r_N$ is minimax-rate-optimal in a non-contaminated setting, th…
We observe $(X_i,Y_i)_{i=1}^n$ where the $Y_i$'s are real valued outputs and the $X_i$'s are $m\times T$ matrices. We observe a new entry $X$ and we want to predict the output $Y$ associated with it. We focus on the high-dimensional setting, where $m T \gg n$. This includes the matrix completion problem with noise, as well as other problems. We consider linear prediction procedures based on different penalizations, involving a mixture of several norms: the nuclear norm, the Frobenius norm and the $\ell_1$-norm. For these procedures, we prove sharp oracle inequalities, using a statistical learn…
We consider a stationary linear AR($p$) model with observations subject to gross errors (outliers). The autoregression parameters are unknown as well as the distribution and moments of innoovations. The distribution of outliers $Π$ is unknown and arbitrary, their intensity is $γn^{-1/2}$ with an unknown $γ$, $n$ is the sample size. The autoregression parameters are estimated by any estimator which is $n^{1/2}$-consistent uniformly in $γ\leq Γ<\infty$. Using the residuals from the estimated autoregression, we construct a kind of empirical distribution function (e.d.f.), which is a counterpart o…
Multivariate Gaussian is often used as a first approximation to the distribution of high-dimensional data. Determining the parameters of this distribution under various constraints is a widely studied problem in statistics, and is often considered as a prototype for testing new algorithms or theoretical frameworks. In this paper, we develop a nonasymptotic approach to the problem of estimating the parameters of a multivariate Gaussian distribution when data are corrupted by outliers. We propose an estimator---efficiently computable by solving a convex program---that robustly estimates the popu…
This paper considers inference in a linear regression model with random right censoring and outliers. The number of outliers can grow with the sample size while their proportion goes to zero. The model is semiparametric and we make only very mild assumptions on the distribution of the error term, contrary to most other existing approaches in the literature. We propose to penalize the estimator proposed by Stute for censored linear regression by the l1-norm. We derive rates of convergence and establish asymptotic normality of the estimator of the regression coefficients. Our estimator has the s…
This paper focuses on exploring the sparsity of the inverse covariance matrix $\bSigma^{-1}$, or the precision matrix. We form blocks of parameters based on each off-diagonal band of the Cholesky factor from its modified Cholesky decomposition, and penalize each block of parameters using the $L_2$-norm instead of individual elements. We develop a one-step estimator, and prove an oracle property which consists of a notion of block sign-consistency and asymptotic normality. In particular, provided the initial estimator of the Cholesky factor is good enough and the true Cholesky has finite number…
Linear regression with normally distributed errors - including particular cases such as ANOVA, Student's t-test or location-scale inference - is a widely used statistical procedure. In this case the ordinary least squares estimator possesses remarkable properties but is very sensitive to outliers. Several robust alternatives have been proposed, but there is still significant room for improvement. This paper thus proposes an original method of estimation that offers the best efficiency simultaneously in the absence and the presence of outliers, both for the estimation of the regression coeffici…
In this paper the integer-valued autoregressive model of order one, contaminated with additive or innovational outliers is studied in some detail. Moreover, parameter estimation is also addressed. Supposing that the time points of the outliers are known but their sizes are unknown, we prove that the Conditional Least Squares (CLS) estimators of the offspring and innovation means are strongly consistent. In contrast, however, the CLS estimators of the outliers' sizes are not strongly consistent, although they converge to a random limit with probability 1. This random limit depends on the values…
We propose a simple multiple outlier identification method for parametric location-scale and shape-scale models when the number of possible outliers is not specified. The method is based on a result giving asymptotic properties of extreme z-scores. Robust estimators of model parameters are used defining z-scores. An extensive simulation study was done for comparing of the proposed method with existing methods. For the normal family, the method is compared with the well known Davies-Gather, Rosner's, Hawking's and Bolshev's multiple outlier identification methods. The choice of an upper limit f…
A large dimensional characterization of robust M-estimators of covariance (or scatter) is provided under the assumption that the dataset comprises independent (essentially Gaussian) legitimate samples as well as arbitrary deterministic samples, referred to as outliers. Building upon recent random matrix advances in the area of robust statistics, we specifically show that the so-called Maronna M-estimator of scatter asymptotically behaves similar to well-known random matrices when the population and sample sizes grow together to infinity. The introduction of outliers leads the robust estimator …
