We develop a constructive approach for $\ell_0$-penalized estimation in the sparse accelerated failure time (AFT) model with high-dimensional covariates. Our proposed method is based on Stute's weighted least squares criterion combined with $\ell_0$-penalization. This method is a computational algorithm that generates a sequence of solutions iteratively, based on active sets derived from primal and dual information and root finding according to the KKT conditions. We refer to the proposed method as AFT-SDAR (for support detection and root finding). An important aspect of our theoretical results is that we directly concern the sequence of solutions generated based on the AFT-SDAR algorithm. We prove that the estimation errors of the solution sequence decay exponentially to the optimal error bound with high probability, as long as the covariate matrix satisfies a mild regularity condition which is necessary and sufficient for model identification even in the setting of high-dimensional linear regression. We also proposed an adaptive version of AFT-SDAR, or AFT-ASDAR, which determines the support size of the estimated coefficient in a data-driven fashion. We conduct simulation studies to demonstrate the superior performance of the proposed method over the lasso and MCP in terms of accuracy and speed. We also apply the proposed method to a real data set to illustrate its application.
This paper is concerned with the column $\ell_{2,0}$-regularized factorization model of low-rank matrix recovery problems and its computation. The column $\ell_{2,0}$-norm of factor matrices is introduced to promote column sparsity of factors and low-rank solutions. For this nonconvex discontinuous optimization problem, we develop an alternating majorization-minimization (AMM) method with extrapolation, and a hybrid AMM in which a majorized alternating proximal method is proposed to seek an initial factor pair with less nonzero columns and the AMM with extrapolation is then employed to minimiz…
This paper is concerned with the factorization form of the rank regularized loss minimization problem. To cater for the scenario in which only a coarse estimation is available for the rank of the true matrix, an $\ell_{2,0}$-norm regularized term is added to the factored loss function to reduce the rank adaptively; and account for the ambiguities in the factorization, a balanced term is then introduced. For the least squares loss, under a restricted condition number assumption on the sampling operator, we establish the KL property of exponent $1/2$ of the nonsmooth factored composite function …
In this paper, we analyse the recovery properties of nonconvex regularized $M$-estimators, under the assumption that the true parameter is of soft sparsity. In the statistical aspect, we establish the recovery bound for any stationary point of the nonconvex regularized $M$-estimator, under restricted strong convexity and some regularity conditions on the loss function and the regularizer, respectively. In the algorithmic aspect, we slightly decompose the objective function and then solve the nonconvex optimization problem via the proximal gradient method, which is proved to achieve a linear co…
We develop a primal dual active set with continuation algorithm for solving the \ell^0-regularized least-squares problem that frequently arises in compressed sensing. The algorithm couples the the primal dual active set method with a continuation strategy on the regularization parameter. At each inner iteration, it first identifies the active set from both primal and dual variables, and then updates the primal variable by solving a (typically small) least-squares problem defined on the active set, from which the dual variable can be updated explicitly. Under certain conditions on the sensing m…
Many state-of-the-art machine learning models such as deep neural networks have recently shown to be vulnerable to adversarial perturbations, especially in classification tasks. Motivated by adversarial machine learning, in this paper we investigate the robustness of sparse regression models with strongly correlated covariates to adversarially designed measurement noises. Specifically, we consider the family of ordered weighted $\ell_1$ (OWL) regularized regression methods and study the case of OSCAR (octagonal shrinkage clustering algorithm for regression) in the adversarial setting. Under a …
In this paper we consider the problem of grouped variable selection in high-dimensional regression using $\ell_1-\ell_q$ regularization ($1\leq q \leq \infty$), which can be viewed as a natural generalization of the $\ell_1-\ell_2$ regularization (the group Lasso). The key condition is that the dimensionality $p_n$ can increase much faster than the sample size $n$, i.e. $p_n \gg n$ (in our case $p_n$ is the number of groups), but the number of relevant groups is small. The main conclusion is that many good properties from $\ell_1-$regularization (Lasso) naturally carry on to the $\ell_1-\ell_q…
Recently, Mahoney and Orecchia demonstrated that popular diffusion-based procedures to compute a quick \emph{approximation} to the first nontrivial eigenvector of a data graph Laplacian \emph{exactly} solve certain regularized Semi-Definite Programs (SDPs). In this paper, we extend that result by providing a statistical interpretation of their approximation procedure. Our interpretation will be analogous to the manner in which $\ell_2$-regularized or $\ell_1$-regularized $\ell_2$-regression (often called Ridge regression and Lasso regression, respectively) can be interpreted in terms of a Gaus…
Variable selection properties of procedures utilizing penalized-likelihood estimates is a central topic in the study of high dimensional linear regression problems. Existing literature emphasizes the quality of ranking of the variables by such procedures as reflected in the receiver operating characteristic curve or in prediction performance. Specifically, recent works have harnessed modern theory of approximate message-passing (AMP) to obtain, in a particular setting, exact asymptotic predictions of the type I-type II error tradeoff for selection procedures that rely on $\ell_{p}$-regularized…
We study the effect of norm based regularization on the size of coresets for regression problems. Specifically, given a matrix $ \mathbf{A} \in {\mathbb{R}}^{n \times d}$ with $n\gg d$ and a vector $\mathbf{b} \in \mathbb{R} ^ n $ and $λ> 0$, we analyze the size of coresets for regularized versions of regression of the form $\|\mathbf{Ax}-\mathbf{b}\|_p^r + λ\|{\mathbf{x}}\|_q^s$ . Prior work has shown that for ridge regression (where $p,q,r,s=2$) we can obtain a coreset that is smaller than the coreset for the unregularized counterpart i.e. least squares regression (Avron et al). We show that…
This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian for the loss function, we derive an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, under the noisy and full sample setting we establish its KL property of exponent $1/2$ on its global minimizer set, and under the noisy and partial sample s…
We consider a discrete optimization formulation for learning sparse classifiers, where the outcome depends upon a linear combination of a small subset of features. Recent work has shown that mixed integer programming (MIP) can be used to solve (to optimality) $\ell_0$-regularized regression problems at scales much larger than what was conventionally considered possible. Despite their usefulness, MIP-based global optimization approaches are significantly slower compared to the relatively mature algorithms for $\ell_1$-regularization and heuristics for nonconvex regularized problems. We aim to b…
Consider the use of $\ell_{1}/\ell_{\infty}$-regularized regression for joint estimation of a $\pdim \times \numreg$ matrix of regression coefficients. We analyze the high-dimensional scaling of $\ell_1/\ell_\infty$-regularized quadratic programming, considering both consistency in $\ell_\infty$-norm, and variable selection. We begin by establishing bounds on the $\ell_\infty$-error as well sufficient conditions for exact variable selection for fixed and random designs. Our second set of results applies to $\numreg = 2$ linear regression problems with standard Gaussian designs whose supports o…
We provide novel theoretical results regarding local optima of regularized $M$-estimators, allowing for nonconvexity in both loss and penalty functions. Under restricted strong convexity on the loss and suitable regularity conditions on the penalty, we prove that \emph{any stationary point} of the composite objective function will lie within statistical precision of the underlying parameter vector. Our theory covers many nonconvex objective functions of interest, including the corrected Lasso for errors-in-variables linear models; regression for generalized linear models with nonconvex penalti…
This article develops a general theory for minimum norm interpolating estimators and regularized empirical risk minimizers (RERM) in linear models in the presence of additive, potentially adversarial, errors. In particular, no conditions on the errors are imposed. A quantitative bound for the prediction error is given, relating it to the Rademacher complexity of the covariates, the norm of the minimum norm interpolator of the errors and the size of the subdifferential around the true parameter. The general theory is illustrated for Gaussian features and several norms: The $\ell_1$, $\ell_2$, g…
Elliptical Regularized Hotelling Testing for High Dimensional Data
School of Statistics and Data Science, Nankai University（南开大学统计与数据科学学院）；Department of Mathematics, Hong Kong Baptist University（香港浸会大学数学系）；Advanced Institute of Natural Sciences, Beijing Normal University（北京师范大学前沿交叉科学研究院）
Feature selection is a standard approach to understanding and modeling high-dimensional classification data, but the corresponding statistical methods hinge on tuning parameters that are difficult to calibrate. In particular, existing calibration schemes in the logistic regression framework lack any finite sample guarantees. In this paper, we introduce a novel calibration scheme for $\ell_1$-penalized logistic regression. It is based on simple tests along the tuning parameter path and is equipped with optimal guarantees for feature selection. It is also amenable to easy and efficient implement…
Graphical models are widely used to model stochastic dependences among large collections of variables. We introduce a new method of estimating undirected conditional independence graphs based on the score matching loss, introduced by Hyvarinen (2005), and subsequently extended in Hyvarinen (2007). The regularized score matching method we propose applies to settings with continuous observations and allows for computationally efficient treatment of possibly non-Gaussian exponential family models. In the well-explored Gaussian setting, regularized score matching avoids issues of asymmetry that ar…
Many scientific and economic problems involve the analysis of high-dimensional time series datasets. However, theoretical studies in high-dimensional statistics to date rely primarily on the assumption of independent and identically distributed (i.i.d.) samples. In this work, we focus on stable Gaussian processes and investigate the theoretical properties of $\ell _1$-regularized estimates in two important statistical problems in the context of high-dimensional time series: (a) stochastic regression with serially correlated errors and (b) transition matrix estimation in vector autoregressive (…
We consider inference about a scalar parameter under a non-parametric model based on a one-step estimator computed as a plug in estimator plus the empirical mean of an estimator of the parameter's influence function. We focus on a class of parameters that have influence function which depends on two infinite dimensional nuisance functions and such that the bias of the one-step estimator of the parameter of interest is the expectation of the product of the estimation errors of the two nuisance functions. Our class includes many important treatment effect contrasts of interest in causal inferenc…
