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正则化统计估计的通用路径算法AI 解读

Regularization is widely used in statistics and machine learning to prevent overfitting and gear solution toward prior information. In general, a regularized estimation problem minimizes the sum of a loss function and a penalty term. The penalty term is usually weighted by a tuning parameter and encourages certain constraints on the parameters to be estimated. Particular choices of constraints lead to the popular lasso, fused-lasso, and other generalized ℓ1 penalized…

基于测量误差模型选择似然的非参数分类变量选择AI 解读

Using the relationships among ridge regression, LASSO estimation, and measurement error attenuation as motivation, a new measurement-error-model-based approach to variable selection is developed. After describing the approach in the familiar context of linear regression, we apply it to the problem of variable selection in nonparametric classification, resulting in a new kernel-based classifier with LASSO-like shrinkage and variable-selection properties. Finite-sample…

动态基因调控网络建模的稀疏可加常微分方程AI 解读

The gene regulation network (GRN) is a high-dimensional complex system, which can be represented by various mathematical or statistical models. The ordinary differential equation (ODE) model is one of the popular dynamic GRN models. High-dimensional linear ODE models have been proposed to identify GRNs, but with a limitation of the linear regulation effect assumption. In this article, we propose a sparse additive ODE (SA-ODE) model, coupled with ODE estimation methods and…

评论AI 解读

Financial data, in particular in the context of managing portfolios and quantifying their risk, come in the form of high-dimensional time series. The evolution of the coordinate processes, represen...

高斯过程混合模型的核平滑估计AI 解读

When the functional data are not homogeneous, e.g., there exist multiple classes of functional curves in the dataset, traditional estimation methods may fail. In this paper, we propose a new estimation procedure for the Mixture of Gaussian Processes, to incorporate both functional and inhomogeneous properties of the data. Our method can be viewed as a natural extension of high-dimensional normal mixtures. However, the key difference is that smoothed structures are imposed…

We propose robust methods for inference about the effect of a treatment variable on a scalar outcome in the presence of very many regressors in a model with possibly non-Gaussian and heteroscedastic disturbances. We allow for the number of regressors to be larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small…

采用收缩与扩散先验的贝叶斯变量选择AI 解读

We consider a Bayesian approach to variable selection in the presence of high dimensional covariates based on a hierarchical model that places prior distributions on the regression coefficients as well as on the model space. We adopt the well-known spike and slab Gaussian priors with a distinct feature, that is, the prior variances depend on the sample size through which appropriate shrinkage can be achieved. We show the strong selection consistency of the proposed method in…

Subspace clustering refers to the task of finding a multi-subspace representation that best fits a collection of points taken from a high-dimensional space. This paper introduces an algorithm inspired by sparse subspace clustering (SSC) [In IEEE Conference on Computer Vision and Pattern Recognition, CVPR (2009) 2790–2797] to cluster noisy data, and develops some novel theory demonstrating its correctness. In particular, the theory uses ideas from geometric functional…

协变量维数增长时相关数据的广义可加部分线性模型估计与选择AI 解读

We propose generalized additive partial linear models for complex data which allow one to capture nonlinear patterns of some covariates, in the presence of linear components. The proposed method improves estimation efficiency and increases statistical power for correlated data through incorporating the correlation information. A unique feature of the proposed method is its capability of handling model selection in cases where it is difficult to specify the likelihood…

非参数回归中基于平方根 Lasso 的枢轴估计AI 解读

We propose a self-tuning $\sqrt{\mathrm {Lasso}} $ method that simultaneously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity and (drastic) non-Gaussianity of the noise. In addition, our analysis allows for badly behaved designs, for example, perfectly collinear regressors, and generates sharp bounds even in extreme cases, such as the infinite variance case and the noiseless case, in…

对《lasso 的显著性检验》讨论的回复AI 解读

We would like to thank the Editors and referees for their considerable efforts that improved our paper, and all of the discussants for their feedback, and their thoughtful and stimulating comments.Linear models are central in applied statistics, and inference for adaptive linear modeling is an important active area of research.Our paper is clearly not the last word on the subject.Several of the discussants introduce novel proposals for this problem; in fact, many of the…

讨论：《Lasso 的显著性检验》AI 解读

The paper by Lockhart, Taylor, Tibshirani and Tibshirani (LTTT) is an important advancement in our understanding of inference for high-dimensional regression.The paper is a tour de force, bringing together an impressive array of results, culminating in a set of very satisfying convergence results.The fact that the test statistic automatically balances the effect of shrinkage and the effect of adaptive variable selection is remarkable.The authors make very strong…

lasso的显著性检验AI 解读

In the sparse linear regression setting, we consider testing the significance of the predictor variable that enters the current lasso model, in the sequence of models visited along the lasso solution path. We propose a simple test statistic based on lasso fitted values, called the covariance test statistic, and show that when the true model is linear, this statistic has an $\operatorname{Exp}(1)$ asymptotic distribution under the null hypothesis (the null being that all…

基于 lasso 惩罚 D-trace 损失的稀疏精度矩阵估计AI 解读

We introduce a constrained empirical loss minimization framework for estimating high-dimensional sparse precision matrices and propose a new loss function, called the D-trace loss, for that purpose. A novel sparse precision matrix estimator is defined as the minimizer of the lasso penalized D-trace loss under a positive-definiteness constraint. Under a new irrepresentability condition, the lasso penalized D-trace estimator is shown to have the sparse recovery property.…

改进的子集回归AI 解读

This paper studies the relationship between model fitting and screening performance to find efficient screening methods for high-dimensional linear regression models. Under a sparsity assumption we show in a general asymptotic setting that a subset that includes the true submodel always yields a smaller residual sum of squares than those that do not. To seek such a subset, we consider the optimization problem associated with best subset regression. An em algorithm, known as…

正则化矩阵回归AI 解读

Summary Modern technologies are producing a wealth of data with complex structures. For instance, in two-dimensional digital imaging, flow cytometry and electroencephalography, matrix-type covariates frequently arise when measurements are obtained for each combination of two underlying variables. To address scientific questions arising from those data, new regression methods that take matrices as covariates are needed, and sparsity or other forms of regularization are…

变量相关情形下高维均值的两样本检验AI 解读

Summary The paper considers in the high dimensional setting a canonical testing problem in multivariate analysis, namely testing the equality of two mean vectors. We introduce a new test statistic that is based on a linear transformation of the data by the precision matrix which incorporates the correlations between the variables. The limiting null distribution of the test statistic and the power of the test are analysed. It is shown that the test is particularly powerful…

用于多类别逆协方差估计的联合图形套索AI 解读

Summary We consider the problem of estimating multiple related Gaussian graphical models from a high dimensional data set with observations belonging to distinct classes. We propose the joint graphical lasso, which borrows strength across the classes to estimate multiple graphical models that share certain characteristics, such as the locations or weights of non-zero edges. Our approach is based on maximizing a penalized log-likelihood. We employ generalized fused lasso or…

稀疏情形下用于变量选择的信息准则AI 解读

The optimization of an information criterion in a variable selection procedure leads to an additional bias, which can be substantial for sparse, high-dimensional data. One can compensate for the bias by applying shrinkage while estimating within the selected models. This paper presents modified information criteria for use in variable selection and estimation without shrinkage. The analysis motivating the modified criteria follows two routes. The first, which we explore for…

联合 L1 与凹惩罚正则化的渐近性质AI 解读

Two important goals of high-dimensional modelling are prediction and variable selection. In this article, we consider regularization with combined L1 and concave penalties, and study the sampling properties of the global optimum of the suggested method in ultrahigh-dimensional settings. The L1 penalty provides the minimum regularization needed for removing noise variables in order to achieve oracle prediction risk, while a concave penalty imposes additional regularization to…

基于多元符号的高维球形性检验AI 解读

This article concerns tests for sphericity in cases where the data dimension is larger than the sample size. The existing multivariate sign-based procedure (Hallin & Paindaveine, 2006) for sphericity is not robust with respect to high dimensionality, producing tests with Type I error rates that are much larger than the nominal levels. This is mainly due to bias from estimating the location parameter. We develop a correction that makes the existing test statistic robust with…

通过趋势滤波进行自适应分段多项式估计AI 解读

We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339–360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute $k$th order discrete derivatives over the input points. Perhaps not surprisingly, trend filtering estimates appear to have the structure of $k$th degree spline functions, with adaptively chosen knot points (we say…

高维信号检测：多尖峰情形AI 解读

This paper applies Le Cam’s asymptotic theory of statistical experiments to the signal detection problem in high dimension. We consider the problem of testing the null hypothesis of sphericity of a high-dimensional covariance matrix against an alternative of (unspecified) multiple symmetry-breaking directions (multispiked alternatives). Simple analytical expressions for the Gaussian asymptotic power envelope and the asymptotic powers of previously proposed tests are derived.…

自适应稳健变量选择AI 解读

Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weighted $L_1$-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the $L_1$-penalty. In the ultra-high dimensional setting, where the dimensionality can grow…