Ridge estimators regularize the squared Euclidean lengths of parameters. Such estimators are mathematically and computationally attractive but involve tuning parameters that can be difficult to calibrate. In this paper, we show that ridge estimators can be modified such that tuning parameters can be avoided altogether. We also show that these modified versions can improve on the empirical prediction accuracies of standard ridge estimators combined with cross-validation, and we provide first theoretical guarantees.
Imputation is a popular technique for handling missing data. We consider a nonparametric approach to imputation using the kernel ridge regression technique and propose consistent variance estimation. The proposed variance estimator is based on a linearization approach which employs the entropy method to estimate the density ratio. The root-n consistency of the imputation estimator is established when a Sobolev space is utilized in the kernel ridge regression imputation, which enables us to develop the proposed variance estimator. Synthetic data experiments are presented to confirm our theory.
Modes and ridges of the probability density function behind observed data are useful geometric features. Mode-seeking clustering assigns cluster labels by associating data samples with the nearest modes, and estimation of density ridges enables us to find lower-dimensional structures hidden in data. A key technical challenge both in mode-seeking clustering and density ridge estimation is accurate estimation of the ratios of the first- and second-order density derivatives to the density. A naive approach takes a three-step approach of first estimating the data density, then computing its deriva…
We propose new model selection criteria based on generalized ridge estimators dominating the maximum likelihood estimator under the squared risk and the Kullback-Leibler risk in multivariate linear regression. Our model selection criteria have the following favorite properties: consistency, unbiasedness, uniformly minimum variance. Consistency is proven under an asymptotic structure $\frac{p}{n}\to c$ where $n$ is the sample size and $p$ is the parameter dimension of the response variables. In particular, our proposed class of estimators dominates the maximum likelihood estimator under the squ…
Amazon（亚马逊）；University of Oxford（牛津大学）；University of Pennsylvania（宾夕法尼亚大学）；Oxford University（牛津大学）
Methods for learning from data depend on various types of tuning parameters, such as penalization strength or step size. Since performance can depend strongly on these parameters, it is important to compare classes of estimators-by considering prescribed finite sets of tuning parameters-not just particularly tuned methods. In this work, we investigate classes of methods via the relative performance of the best method in the class. We consider the central problem of linear regression-with a random isotropic ground truth-and investigate the estimation performance of two fundamental methods, grad…
Focusing on a high dimensional linear model $y = Xβ+ ε$ with dependent, non-stationary, and heteroskedastic errors, this paper applies the debiased and threshold ridge regression method that gives a consistent estimator for linear combinations of $β$; and derives a Gaussian approximation theorem for the estimator. Besides, it proposes a dependent wild bootstrap algorithm to construct the estimator's confidence intervals and perform hypothesis testing. Numerical experiments on the proposed estimator and the bootstrap algorithm show that they have favorable finite sample performance. Research on…
We introduce an original method of multidimensional ridge penalization in functional local linear regressions. The nonparametric regression of functional data is extended from its multivariate counterpart, and is known to be sensitive to the choice of $J$, where $J$ is the dimension of the projection subspace of the data. Under multivariate setting, a roughness penalty is helpful for variance reduction. However, among the limited works covering roughness penalty under the functional setting, most only use a single scalar for tuning. Our new approach proposes a class of data-adaptive ridge pena…
For high-dimensional linear regression models, we review and compare several estimators of variances $τ^2$ and $σ^2$ of the random slopes and errors, respectively. These variances relate directly to ridge regression penalty $λ$ and heritability index $h^2$, often used in genetics. Direct and indirect estimators of these, either based on cross-validation (CV) or maximum marginal likelihood (MML), are also discussed. The comparisons include several cases of covariate matrix $\mathbf{X}_{n \times p}$, with $p \gg n$, such as multi-collinear covariates and data-derived ones. In addition, we study …
We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an independent kernel ridge regression estimator for each subset, then averages the local solutions into a global predictor. This partitioning leads to a substantial reduction in computation time versus the standard approach of performing kernel ridge regression on all N samples. Our two main theorems establish that despite the computational speed-up, statistical optimal…
Separation in logistic regression is a common problem causing failure of the iterative estimation process when finding maximum likelihood estimates. Firth's correction (FC) was proposed as a solution, providing estimates also in presence of separation. In this paper we evaluate whether ridge regression (RR) could be considered instead, specifically, if it could reduce the mean squared error (MSE) of coefficient estimates in comparison to FC. In RR the tuning parameter determining the penalty strength is usually obtained by minimizing some measure of the out-of-sample prediction error or inform…
Constructing confidence intervals for the coefficients of high-dimensional sparse linear models remains a challenge, mainly because of the complicated limiting distributions of the widely used estimators, such as the lasso. Several methods have been developed for constructing such intervals. Bootstrap lasso+ols is notable for its technical simplicity, good interpretability, and performance that is comparable with that of other more complicated methods. However, bootstrap lasso+ols depends on the beta-min assumption, a theoretic criterion that is often violated in practice. Thus, we introduce a…
We develop large sample theory including nonparametric confidence regions for $r$-dimensional ridges of probability density functions on $\mathbb{R}^d$, where $1\leq r<d$. We view ridges as the intersections of level sets of some special functions. The vertical variation of the plug-in kernel estimators for these functions constrained on the ridges is used as the measure of maximal deviation for ridge estimation. Our confidence regions for the ridges are determined by the asymptotic distribution of this maximal deviation, which is established by utilizing the extreme value distribution of nons…
We propose statistical inferential procedures for panel data models with interactive fixed effects in a kernel ridge regression framework.Compared with traditional sieve methods, our method is automatic in the sense that it does not require the choice of basis functions and truncation parameters.Model complexity is controlled by a continuous regularization parameter which can be automatically selected by generalized cross validation. Based on empirical processes theory and functional analysis tools, we derive joint asymptotic distributions for the estimators in the heterogeneous setting. These…
The large sample theory of estimators for density modes is well understood. In this paper we consider density ridges, which are a higher-dimensional extension of modes. Modes correspond to zero-dimensional, local high-density regions in point clouds. Density ridges correspond to $s$-dimensional, local high-density regions in point clouds. We establish three main results. First we show that under appropriate regularity conditions, the local variation of the estimated ridge can be approximated by an empirical process. Second, we show that the distribution of the estimated ridge converges to a Ga…
Regularization is an essential element of virtually all kernel methods for nonparametric regression problems. A critical factor in the effectiveness of a given kernel method is the type of regularization that is employed. This article compares and contrasts members from a general class of regularization techniques, which notably includes ridge regression and principal component regression. We derive an explicit finite-sample risk bound for regularization-based estimators that simultaneously accounts for (i) the structure of the ambient function space, (ii) the regularity of the true regression…
Kernel methods for deconvolution have attractive features, and prevail in the literature. However, they have disadvantages, which include the fact that they are usually suitable only for cases where the error distribution is infinitely supported and its characteristic function does not ever vanish. Even in these settings, optimal convergence rates are achieved by kernel estimators only when the kernel is chosen to adapt to the unknown smoothness of the target distribution. In this paper we suggest alternative ridge methods, not involving kernels in any way. We show that ridge methods (a) do no…
Beta regression model is useful in the analysis of bounded continuous outcomes such as proportions. It is well known that for any regression model, the presence of multicollinearity leads to poor performance of the maximum likelihood estimators. The ridge type estimators have been proposed to alleviate the adverse effects of the multicollinearity. Furthermore, when some of the predictors have insignificant or weak effects on the outcomes, it is desired to recover as much information as possible from these predictors instead of discarding them all together. In this paper we proposed ridge type …
It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, in the context of biased shrinkage ridge estimation, Chang (2015) introduced an almost unbiased Liu estimator in the logistic regression model. Making use of his approach, when some prior knowledge in the form of linear restrictions are also available, we introduce a restricted almost unbiased Liu estimator in the logistic regression model. Statistical properties of this newly defined estimator are derived and some comparison result are also pr…
University of Washington（华盛顿大学）；AWS AI Labs（AWS人工智能实验室）
Determining causal relationship between high dimensional observations are among the most important tasks in scientific discoveries. In this paper, we revisited the \emph{linear trace method}, a technique proposed in~\citep{janzing2009telling,zscheischler2011testing} to infer the causal direction between two random variables of high dimensions. We strengthen the existing results significantly by providing an improved tail analysis in addition to extending the results to nonlinear trace functionals with sharper confidence bounds under certain distributional assumptions. We obtain our results by …
University of Macau（澳门大学）；Zhuhai UM Science & Technology Research Institute（珠海澳大科技研究院）；Peking University（北京大学）
There has been a surge of interest in developing robust estimators for models with heavy-tailed and bounded variance data in statistics and machine learning, while few works impose unbounded variance. This paper proposes two type of robust estimators, the ridge log-truncated M-estimator and the elastic net log-truncated M-estimator. The first estimator is applied to convex regressions such as quantile regression and generalized linear models, while the other one is applied to high dimensional non-convex learning problems such as regressions via deep neural networks. Simulations and real data a…
Ridge estimator is an alternative to ordinary least square estimator when there is multicollinearity problem. There are many proposed estimators in literature. In this paper, we propose new estimators which are modifications of the estimator suggested by Lawless and Wang (1976). A Monte Carlo experiment has been conducted for the comparison of the performances of the estimators. Mean squared error (MSE) is used as a performance criterion. The benefits of new estimators are illustrated using two real datasets. According to both simulation results and applications, our new estimators have better…
We study ridge estimation of the precision matrix in the high-dimensional setting where the number of variables is large relative to the sample size. We first review two archetypal ridge estimators and note that their utilized penalties do not coincide with common ridge penalties. Subsequently, starting from a common ridge penalty, analytic expressions are derived for two alternative ridge estimators of the precision matrix. The alternative estimators are compared to the archetypes with regard to eigenvalue shrinkage and risk. The alternatives are also compared to the graphical lasso within th…
Bayesian methodologies prioritising accurate associations above sparsity in Gaussian graphical model (GGM) estimation remain relatively scarce in scientific literature. It is well accepted that the $\ell_2$ penalty enjoys a smaller computational footprint in GGM estimation, whilst the $\ell_1$ penalty encourages sparsity in the estimand. The Bayesian adaptive graphical lasso prior is used as a departure point in the formulation of a computationally efficient graphical ridge-type prior for events where accurate associations are prioritised over sparse representations. A novel block Gibbs sample…
With a focus on linear models with smooth functional covariates, we propose a penalization framework (SACR) based on the nonzero centered ridge, where the center of the penalty is optimally reweighted in a supervised way, starting from the ordinary ridge solution as the initial centerfunction. In particular, we introduce a convex formulation that jointly estimates the model's coefficients and the weight function, with a roughness penalty on the centerfunction and constraints on the weights in order to recover a possibly smooth and/or sparse solution. This allows for a non-iterative and continu…
We introduce the concept of coverage risk as an error measure for density ridge estimation. The coverage risk generalizes the mean integrated square error to set estimation. We propose two risk estimators for the coverage risk and we show that we can select tuning parameters by minimizing the estimated risk. We study the rate of convergence for coverage risk and prove consistency of the risk estimators. We apply our method to three simulated datasets and to cosmology data. In all the examples, the proposed method successfully recover the underlying density structure.
Marginal association summary statistics have attracted great attention in statistical genetics, mainly because the primary results of most genome-wide association studies (GWAS) are produced by marginal screening. In this paper, we study the prediction accuracy of marginal estimator in dense (or sparsity free) high-dimensional settings with $(n,p,m) \to \infty$, $m/n \to γ\in (0,\infty)$, and $p/n \to ω\in (0,\infty)$. We consider a general correlation structure among the $p$ features and allow an unknown subset $m$ of them to be signals. As the marginal estimator can be viewed as a ridge esti…
Subsampling is a popular approach to alleviating the computational burden for analyzing massive datasets. Recent efforts have been devoted to various statistical models without explicit regularization. In this paper, we develop an efficient subsampling procedure for the large sample linear ridge regression. In contrast to the ordinary least square estimator, the introduction of the ridge penalty leads to a subtle trade-off between bias and variance. We first investigate the asymptotic properties of the subsampling estimator and then propose to minimize the asymptotic-mean-squared-error criteri…
We consider the problem of jointly estimating multiple inverse covariance matrices from high-dimensional data consisting of distinct classes. An $\ell_2$-penalized maximum likelihood approach is employed. The suggested approach is flexible and generic, incorporating several other $\ell_2$-penalized estimators as special cases. In addition, the approach allows specification of target matrices through which prior knowledge may be incorporated and which can stabilize the estimation procedure in high-dimensional settings. The result is a targeted fused ridge estimator that is of use when the preci…
Unlike the ordinary least-squares (OLS) estimator for the linear model, a ridge regression linear model provides coefficient estimates via shrinkage, usually with improved mean-square and prediction error. This is true especially when the observed design matrix is ill-conditioned or singular, either as a result of highly-correlated covariates or the number of covariates exceeding the sample size. This paper introduces novel and fast marginal maximum likelihood (MML) algorithms for estimating the shrinkage parameter(s) for the Bayesian ridge and power ridge regression models, and an automatic p…
This paper characterizes the conditional distribution properties of the finite sample ridge regression estimator and uses that result to evaluate total regression and generalization errors that incorporate the inaccuracies committed at the time of parameter estimation. The paper provides explicit formulas for those errors. Unlike other classical references in this setup, our results take place in a fully singular setup that does not assume the existence of a solution for the non-regularized regression problem. In exchange, we invoke a conditional homoscedasticity hypothesis on the regularized …
Imputation and propensity score weighting are two popular techniques for handling missing data. We address these problems using the regularized M-estimation techniques in the reproducing kernel Hilbert space. Specifically, we first use the kernel ridge regression to develop imputation for handling item nonresponse. While this nonparametric approach is potentially promising for imputation, its statistical properties are not investigated in the literature. Under some conditions on the order of the tuning parameter, we first establish the root-$n$ consistency of the kernel ridge regression imputa…
