A generalized elastic net regularization with smoothed $$\ell _{q}$$ penalty for sparse vector recovery

In this paper, we propose an iterative algorithm for solving the generalized elastic net regularization problem with smoothed \(\ell _{q} (0<q \le 1)\) penalty for recovering sparse vectors. We prove the convergence result of the algorithm based on the algebraic method. Under certain conditions, we show that the iterative solutions converge to a local minimizer of the generalized elastic net regularization problem and we also present an error bound. Theoretical analysis and numerical results show that the proposed algorithm is promising.     

An augmented Lagrangian approach for sparse principal component analysis

Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in the literature (Cadima and Jolliffe in J Appl Stat 22:203–214, 1995; d’Aspremont et?al. in J Mach Learn Res 9:1269–1294, 2008; d’Aspremont et?al. SIAM Rev 49:434–448, 2007; Jolliffe in J Appl Stat 22:29–35, 1995; Journée et?al. in J Mach Learn Res 11:517–553, 2010; Jolliffe et?al. in J Comput Graph Stat 12:531–547, 2003; Moghaddam et?al. in Advances in neural information processing systems 18:915–922, MIT Press, Cambridge, 2006; Shen and Huang in J Multivar Anal 99(6):1015–1034, 2008; Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006). Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic (Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006), Pitprops (Jeffers in Appl Stat 16:225–236, 1967), and gene expression data (Chin et?al in Cancer Cell 10:529C–541C, 2006), respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors. Moreover, the experiments on random data show that our method is capable of solving large-scale problems within a reasonable amount of time.     

Autocovariance Function Estimation via Penalized Regression

The work revisits the autocovariance function estimation, a fundamental problem in statistical inference for time series. We convert the function estimation problem into constrained penalized regression with a generalized penalty that provides us with flexible and accurate estimation, and study the asymptotic properties of the proposed estimator. In case of a nonzero mean time series, we apply a penalized regression technique to a differenced time series, which does not require a separate detrending procedure. In penalized regression, selection of tuning parameters is critical and we propose four different data-driven criteria to determine them. A simulation study shows effectiveness of the tuning parameter selection and that the proposed approach is superior to three existing methods. We also briefly discuss the extension of the proposed approach to interval-valued time series. Supplementary materials for this article are available online.     

Robust and efficient variable selection for semiparametric partially linear varying coefficient model based on modal regression

Semiparametric partially linear varying coefficient models (SPLVCM) are frequently used in statistical modeling. With high-dimensional covariates both in parametric and nonparametric part for SPLVCM, sparse modeling is often considered in practice. In this paper, we propose a new estimation and variable selection procedure based on modal regression, where the nonparametric functions are approximated by $B$ -spline basis. The outstanding merit of the proposed variable selection procedure is that it can achieve both robustness and efficiency by introducing an additional tuning parameter (i.e., bandwidth $h$ ). Its oracle property is also established for both the parametric and nonparametric part. Moreover, we give the data-driven bandwidth selection method and propose an EM-type algorithm for the proposed method. Monte Carlo simulation study and real data example are conducted to examine the finite sample performance of the proposed method. Both the simulation results and real data analysis confirm that the newly proposed method works very well.     

Group-Wise Principal Component Analysis for Exploratory Data Analysis

In this article, we propose a new framework for matrix factorization based on principal component analysis (PCA) where sparsity is imposed. The structure to impose sparsity is defined in terms of groups of correlated variables found in correlation matrices or maps. The framework is based on three new contributions: an algorithm to identify the groups of variables in correlation maps, a visualization for the resulting groups, and a matrix factorization. Together with a method to compute correlation maps with minimum noise level, referred to as missing-data for exploratory data analysis (MEDA), these three contributions constitute a complete matrix factorization framework. Two real examples are used to illustrate the approach and compare it with PCA, sparse PCA, and structured sparse PCA. Supplementary materials for this article are available online.     

Sparse Distance Weighted Discrimination

Distance weighted discrimination (DWD) was originally proposed to handle the data piling issue in the support vector machine. In this article, we consider the sparse penalized DWD for high-dimensional classification. The state-of-the-art algorithm for solving the standard DWD is based on second-order cone programming, however such an algorithm does not work well for the sparse penalized DWD with high-dimensional data. To overcome the challenging computation difficulty, we develop a very efficient algorithm to compute the solution path of the sparse DWD at a given fine grid of regularization parameters. We implement the algorithm in a publicly available R package sdwd. We conduct extensive numerical experiments to demonstrate the computational efficiency and classification performance of our method.     

Sparse Multivariate Regression With Covariance Estimation

We propose a procedure for constructing a sparse estimator of a multivariate regression coefficient matrix that accounts for correlation of the response variables. This method, which we call multivariate regression with covariance estimation (MRCE), involves penalized likelihood with simultaneous estimation of the regression coefficients and the covariance structure. An efficient optimization algorithm and a fast approximation are developed for computing MRCE. Using simulation studies, we show that the proposed method outperforms relevant competitors when the responses are highly correlated. We also apply the new method to a finance example on predicting asset returns. An R-package containing this dataset and code for computing MRCE and its approximation are available online.     

Penalized regression with individual deviance effects

The present work addresses the problem of model estimation and computations for discrete data when some covariates are modeled smoothly using splines. We propose to introduce and explicitly estimate individual deviance effects (one for each observation), constrained by a ridge penalty. This turns out to be an effective way to absorb model excess variation and detect systematic patterns. Large but very sparse systems of penalized likelihood equations have to be solved. We present fast and compact algorithms for fitting, estimation and computation of the effective dimension. Applications to counts, binomial, and survival data illustrate practical use of this model.     

Supervised Sparse and Functional Principal Component Analysis

Principal component analysis (PCA) is an important tool for dimension reduction in multivariate analysis. Regularized PCA methods, such as sparse PCA and functional PCA, have been developed to incorporate special features in many real applications. Sometimes additional variables (referred to as supervision) are measured on the same set of samples, which can potentially drive low-rank structures of the primary data of interest. Classical PCA methods cannot make use of such supervision data. In this article, we propose a supervised sparse and functional principal component (SupSFPC) framework that can incorporate supervision information to recover underlying structures that are more interpretable. The framework unifies and generalizes several existing methods and flexibly adapts to the practical scenarios at hand. The SupSFPC model is formulated in a hierarchical fashion using latent variables. We develop an efficient modified expectation-maximization (EM) algorithm for parameter estimation. We also implement fast data-driven procedures for tuning parameter selection. Our comprehensive simulation and real data examples demonstrate the advantages of SupSFPC. Supplementary materials for this article are available online.     

A-线性Bregman 迭代算法

线性Bregman迭代是Osher和Cai等人最近提出的一种在压缩感知等领域有重要作用的有效算法.本文在矩阵A非满秩情形下,研究了求解下面最优化问题的线性Bregman迭代:min u∈R~M{‖u‖_1:Au+g}给出了一个关于线性Bregman迭代收敛性定理的简化证明,设计了一类A~-线性Bregman迭代算法,并针对A~+情形证明了算法的收敛性.最后,用数值仿真实验验证了本文算法的可行性.     

Spatially Weighted Principal Component Analysis for Imaging Classification

The aim of this article is to develop a supervised dimension-reduction framework, called spatially weighted principal component analysis (SWPCA), for high-dimensional imaging classification. Two main challenges in imaging classification are the high dimensionality of the feature space and the complex spatial structure of imaging data. In SWPCA, we introduce two sets of novel weights, including global and local spatial weights, which enable a selective treatment of individual features and incorporation of the spatial structure of imaging data and class label information. We develop an efficient two-stage iterative SWPCA algorithm and its penalized version along with the associated weight determination. We use both simulation studies and real data analysis to evaluate the finite-sample performance of our SWPCA. The results show that SWPCA outperforms several competing principal component analysis (PCA) methods, such as supervised PCA (SPCA), and other competing methods, such as sparse discriminant analysis (SDA).     

Multilevel quasi-interpolation on a sparse grid with the Gaussian

Motivated by the recent multilevel sparse kernel-based interpolation (MuSIK) algorithm proposed in Georgoulis et al. (SIAM J. Sci. Comput. 35, 815–832, 2013), we introduce the new quasi-multilevel sparse interpolation with kernels (Q-MuSIK) via the combination technique. The Q-MuSIK scheme achieves better convergence and run time when compared with classical quasi-interpolation. Also, the Q-MuSIK algorithm is generally superior to the MuSIK methods in terms of run time in particular in high-dimensional interpolation problems, since there is no need to solve large algebraic systems. We subsequently propose a fast, low complexity, high-dimensional positive-weight quadrature formula based on Q-MuSIKSapproximation of the integrand. We present the results of numerical experimentation for both quasi-interpolation and quadrature in high dimensions.     

An Augmented ADMM Algorithm With Application to the Generalized Lasso Problem

In this article, we present a fast and stable algorithm for solving a class of optimization problems that arise in many statistical estimation procedures, such as sparse fused lasso over a graph, convex clustering, and trend filtering, among others. We propose a so-called augmented alternating direction methods of multipliers (ADMM) algorithm to solve this class of problems. Compared to a standard ADMM algorithm, our proposal significantly reduces the computational cost at each iteration while maintaining roughly the same overall convergence speed. We also consider a new varying penalty scheme for the ADMM algorithm, which could further accelerate the convergence, especially when solving a sequence of problems with tuning parameters of different scales. Extensive numerical experiments on the sparse fused lasso problem show that the proposed algorithm is more efficient than the standard ADMM and two other existing state-of-the-art specialized algorithms. Finally, we discuss a possible extension and some interesting connections to two well-known algorithms. Supplementary materials for the article are available online.     

Numerically stable and efficient algorithm for vector channel parameter and frequency offset estimation

In this paper, we present a novel and numerically efficient algorithm for vector channel and calibration vector estimation, which works when frequency offset error caused by either unstable oscillator or Doppler effect is present in Spread Spectrum antenna system. We propose an estimation algorithm based on Gauss–Seidal algorithm rather than using eigen-decomposition or SVD in computing eigenvalues and eigenvectors at each iteration. The algorithm is based on the two-step procedures, one for estimating both channel and frequency offset and the other for estimating the unknown array gain and phase. Consequently, estimates of the DOAs, the multi-path impulse response of the reference signal source, and the carrier frequency offset as well as the calibration of antenna array are provided. The analytic performance improvement in multiplications number is presented. The performance of the proposed algorithm is investigated by means of computer simulations. Throughout the analytic and computer simulation, we show that the proposed algorithm reduces the number of multiplications by order of one.     

A modified greedy analysis pursuit algorithm for the cosparse analysis model

In the past decade, the sparse representation synthesis model has been deeply researched and widely applied in signal processing. Recently, a cosparse analysis model has been introduced as an interesting alternative to the sparse representation synthesis model. The sparse synthesis model pay attention to non-zero elements in a representation vector x, while the cosparse analysis model focuses on zero elements in the analysis representation vector Ωx. This paper mainly considers the problem of the cosparse analysis model. Based on the greedy analysis pursuit algorithm, by constructing an adaptive weighted matrix W _k?1, we propose a modified greedy analysis pursuit algorithm for the sparse recovery problem when the signal obeys the cosparse model. Using a weighted matrix, we fill the gap between greedy algorithm and relaxation techniques. The standard analysis shows that our algorithm is convergent. We estimate the error bound for solving the cosparse analysis model, and then the presented simulations demonstrate the advantage of the proposed method for the cosparse inverse problem.     

An improved model averaging scheme for logistic regression

Recently, penalized regression methods have attracted much attention in the statistical literature. In this article, we argue that such methods can be improved for the purposes of prediction by utilizing model averaging ideas. We propose a new algorithm that combines penalized regression with model averaging for improved prediction. We also discuss the issue of model selection versus model averaging and propose a diagnostic based on the notion of generalized degrees of freedom. The proposed methods are studied using both simulated and real data.     

Local convergence of generalized Mann iteration

The local convergence of generalized Mann iteration is investigated in the setting of a real Hilbert space. As application, we obtain an algorithm for estimating the local radius of convergence for some known iterative methods. Numerical experiments are presented showing the performances of the proposed algorithm. For a particular case of the Ezquerro-Hernandez method (Ezquerro and Hernandez, J. Complex., 25:343–361: 2009), the proposed procedure gives radii which are very close to or even identical with the best possible ones.     

Model selection and sharp asymptotic minimaxity

We obtain sharp minimax results for estimation of an n-dimensional normal mean under quadratic loss. The estimators are chosen by penalized least squares with a penalty that grows like ck log(n/k), for k equal to the number of nonzero elements in the estimating vector. For a wide range of sparse parameter spaces, we show that the penalized estimator achieves the exact minimax rate with the correct multiplication constant if and only if c equals 2. Our results unify the theory obtained by many other authors for penalized estimation of normal means. In particular we establish that a conjecture by Abramovich et al. (Ann Stat 34:584–653, 2006) is true.     

A Class of Incomplete Orthogonal Factorization Methods. I: Methods and Theories

We present a class of incomplete orthogonal factorization methods based on Givens rotations for large sparse unsymmetric matrices. These methods include: Incomplete Givens Orthogonalization (IGO-method) and its generalisation (GIGO-method), which drop entries from the incomplete orthogonal and upper triangular factors by position; Threshold Incomplete Givens Orthogonalization (TIGO()-method), which drops entries dynamically by their magnitudes; and its generalisation (GTIGO(,p)-method), which drops entries dynamically by both their magnitudes and positions. Theoretical analyses show that these methods can produce a nonsingular sparse incomplete upper triangular factor and either a complete orthogonal factor or a sparse nonsingular incomplete orthogonal factor for a general nonsingular matrix. Therefore, these methods can potentially generate efficient preconditioners for Krylov subspace methods for solving large sparse systems of linear equations. Moreover, the upper triangular factor is an incomplete Cholesky factorization preconditioner for the normal equations matrix from least-squares problems.     

Sparse Steinian Covariance Estimation

We consider a new method for sparse covariance matrix estimation which is motivated by previous results for the so-called Stein-type estimators. Stein proposed a method for regularizing the sample covariance matrix by shrinking together the eigenvalues; the amount of shrinkage is chosen to minimize an unbiased estimate of the risk (UBEOR) under the entropy loss function. The resulting estimator has been shown in simulations to yield significant risk reductions over the maximum likelihood estimator. Our method extends the UBEOR minimization problem by adding an ?₁ penalty on the entries of the estimated covariance matrix, which encourages a sparse estimate. For a multivariate Gaussian distribution, zeros in the covariance matrix correspond to marginal independences between variables. Unlike the ?₁-penalized Gaussian likelihood function, our penalized UBEOR objective is convex and can be minimized via a simple block coordinate descent procedure. We demonstrate via numerical simulations and an analysis of microarray data from breast cancer patients that our proposed method generally outperforms other methods for sparse covariance matrix estimation and can be computed efficiently even in high dimensions.