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格点数据的多元趋势滤波AI 解读

We study a multivariate version of trend filtering, called Kronecker trend filtering or KTF, for the case in which the design points form a lattice in d dimensions. KTF is a natural extension of univariate trend filtering (Int. J. Comput. Vis. 70 (2006) 214–255; SIAM Rev. 51 (2009) 339–360; Ann. Statist. 42 (2014) 285–323), and is defined by minimizing a penalized least squares problem whose penalty term sums the absolute (higher-order) differences of the parameter to be…

高维无岭最小二乘插值中的意外现象AI 解读

Interpolators—estimators that achieve zero training error—have attracted growing attention in machine learning, mainly because state-of-the art neural networks appear to be models of this type. In this paper, we study minimum ℓ2 norm (“ridgeless”) interpolation least squares regression, focusing on the high-dimensional regime in which the number of unknown parameters p is of the same order as the number of samples n. We consider two different models for the feature…

利用非凸融合惩罚建模高维分类数据AI 解读

Abstract We propose a method for estimation in high-dimensional linear models with nominal categorical data. Our estimator, called SCOPE, fuses levels together by making their corresponding coefficients exactly equal. This is achieved using the minimax concave penalty on differences between the order statistics of the coefficients for a categorical variable, thereby clustering the coefficients. We provide an algorithm for exact and efficient computation of the global minimum…

使用 jackknife+ 的预测推断AI 解读

This paper introduces the jackknife+, which is a novel method for constructing predictive confidence intervals. Whereas the jackknife outputs an interval centered at the predicted response of a test point, with the width of the interval determined by the quantiles of leave-one-out residuals, the jackknife+ also uses the leave-one-out predictions at the test point to account for the variability in the fitted regression function. Assuming exchangeable training samples, we…

从固定设计到随机设计回归：偏差方差分解、协方差惩罚与预测误差估计AI 解读

In statistical prediction, classical approaches for model selection and model evaluation based on covariance penalties are still widely used. Most of the literature on this topic is based on what we call the “Fixed-X” assumption, where covariate values are assumed to be nonrandom. By contrast, it is often more reasonable to take a “Random-X” view, where the covariate values are independently drawn for both training and prediction. To study the applicability of covariance…

基于趋势滤波的加性模型AI 解读

We study additive models built with trend filtering, that is, additive models whose components are each regularized by the (discrete) total variation of their $k$th (discrete) derivative, for a chosen integer $k\geq0$. This results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives piecewise constant components, $k=1$ gives piecewise linear, $k=2$ gives piecewise quadratic, etc.). Analogous to its advantages in the univariate case, additive trend filtering…

过度乐观：经 SURE 调参后的表观误差有多大偏差？AI 解读

Nearly all estimators in statistical prediction come with an associated tuning parameter, in one way or another. Common practice, given data, is to choose the tuning parameter value that minimizes a constructed estimate of the prediction error of the estimator; we focus on Stein’s unbiased risk estimator, or SURE, which forms an unbiased estimate of the prediction error by augmenting the observed training error with an estimate of the degrees of freedom of the estimator.…

回归中的分布无关预测推断AI 解读

We develop a general framework for distribution-free predictive inference in regression, using conformal inference. The proposed methodology allows for the construction of a prediction band for the response variable using any estimator of the regression function. The resulting prediction band preserves the consistency properties of the original estimator under standard assumptions, while guaranteeing finite-sample marginal coverage even when these assumptions do not hold. We…

模型选择后的统一渐近推断与 BootstrapAI 解读

Recently, Tibshirani et al. [J. Amer. Statist. Assoc. 111 (2016) 600–620] proposed a method for making inferences about parameters defined by model selection, in a typical regression setting with normally distributed errors. Here, we study the large sample properties of this method, without assuming normality. We prove that the test statistic of Tibshirani et al. (2016) is asymptotically valid, as the number of samples $n$ grows and the dimension $d$ of the regression…

序贯回归程序的精确选择后推断AI 解读

We propose new inference tools for forward stepwise regression, least angle regression, and the lasso. Assuming a Gaussian model for the observation vector y, we first describe a general scheme to perform valid inference after any selection event that can be characterized as y falling into a polyhedral set. This framework allows us to derive conditional (post-selection) hypothesis tests at any step of forward stepwise or least angle regression, or any step along the lasso…

利用 Kac–Rice 公式进行自适应回归推断AI 解读

We derive an exact $p$-value for testing a global null hypothesis in a general adaptive regression setting. Our approach uses the Kac–Rice formula [as described in Random Fields and Geometry (2007) Springer, New York] applied to the problem of maximizing a Gaussian process. The resulting test statistic has a known distribution in finite samples, assuming Gaussian errors. We examine this test statistic in the case of the lasso, group lasso, principal components and matrix…

非参数众数回归AI 解读

Modal regression estimates the local modes of the distribution of $Y$ given $X=x$, instead of the mean, as in the usual regression sense, and can hence reveal important structure missed by usual regression methods. We study a simple nonparametric method for modal regression, based on a kernel density estimate (KDE) of the joint distribution of $Y$ and $X$. We derive asymptotic error bounds for this method, and propose techniques for constructing confidence sets and…

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…

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…

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…

Lasso 问题中的自由度AI 解读

We derive the degrees of freedom of the lasso fit, placing no assumptions on the predictor matrix $X$. Like the well-known result of Zou, Hastie and Tibshirani [Ann. Statist. 35 (2007) 2173–2192], which gives the degrees of freedom of the lasso fit when $X$ has full column rank, we express our result in terms of the active set of a lasso solution. We extend this result to cover the degrees of freedom of the generalized lasso fit for an arbitrary predictor matrix $X$ (and an…

Lasso类问题中剔除预测变量的强规则AI 解读

Summary We consider rules for discarding predictors in lasso regression and related problems, for computational efficiency. El Ghaoui and his colleagues have proposed ‘SAFE’ rules, based on univariate inner products between each predictor and the outcome, which guarantee that a coefficient will be 0 in the solution vector. This provides a reduction in the number of variables that need to be entered into the optimization. We propose strong rules that are very simple and yet…

广义 lasso 的解路径AI 解读

We present a path algorithm for the generalized lasso problem. This problem penalizes the ℓ1 norm of a matrix D times the coefficient vector, and has a wide range of applications, dictated by the choice of D. Our algorithm is based on solving the dual of the generalized lasso, which greatly facilitates computation of the path. For D = I (the usual lasso), we draw a connection between our approach and the well-known LARS algorithm. For an arbitrary D, we derive an unbiased…