我们研究线性模型的调优正则化参数估计器求解问题。首先证明三种已知的最优线性估计器属于更广泛的估计器类别，这类估计器可表述为带约束的加权最小化问题的解。但在许多实际应用中，最优权重通常是未知的，这就引出了一个问题：如何仅利用数据来选择权重？我们提出采用协方差拟合SPICE方法（SPICE即基于迭代协方差拟合的稀疏估计方法）来获取数据自适应权重，并证明由此得到的估计器类别可生成已知正则化估计器的调优版本，例如岭回归（ridge regression）、LASSO和正则化最小绝对偏差（regularized least absolute deviation）。这些理论结果将多个重要估计器统一到了同一框架下。通过大量数值示例，我们还证明了所得调优估计器具有实际应用价值。
英文摘要：
We consider the problem of finding tuned regularized parameter estimators for linear models. We start by showing that three known optimal linear estimators belong to a wider class of estimators that can be formulated as a solution to a weighted and constrained minimization problem. The optimal weights, however, are typically unknown in many applications. This begs the question, how should we choose the weights using only the data? We propose using the covariance fitting SPICE-methodology to obtain data-adaptive weights and show that the resulting class of estimators yields tuned versions of known regularized estimators - such as ridge regression, LASSO, and regularized least absolute deviation. These theoretical results unify several important estimators under a common umbrella. The resulting tuned estimators are also shown to be practically relevant by means of a number of numerical examples.
The maximum likelihood principle is widely used in statistics, and the associated estimators often display good properties. indeed maximum likelihood estimators are guaranteed to be asymptotically efficient under mild conditions. However in some settings, one has too few samples to get a good estimation. It then becomes desirable to take into account prior information about the distribution one wants to estimate. One possible approach is to extend the maximum likelihood principle in a bayesian context, which then becomes a maximum a posteriori estimate; however this requires a distribution mod…
The main theme of this paper is a modification of the likelihood ratio test (LRT) for testing high dimensional covariance matrix. Recently, the correct asymptotic distribution of the LRT for a large-dimensional case (the case $p/n$ approaches to a constant $γ\in (0,1]$) is specified by researchers. The correct procedure is named as corrected LRT. Despite of its correction, the corrected LRT is a function of sample eigenvalues that are suffered from redundant variability from high dimensionality and, subsequently, still does not have full power in differentiating hypotheses on the covariance ma…
Missing data estimation is an important challenge with high-dimensional data arranged in the form of a matrix. Typically this data matrix is transposable, meaning that either the rows, columns or both can be treated as features. To model transposable data, we present a modification of the matrix-variate normal, the mean-restricted matrix-variate normal, in which the rows and columns each have a separate mean vector and covariance matrix. By placing additive penalties on the inverse covariance matrices of the rows and columns, these so-called transposable regularized covariance models allow for…
In this paper, the asymptotic distributions of estimators for the regularized functional canonical correlation and variates of the population are derived. The method is based on the possibility of expressing these regularized quantities as the maximum eigenvalue and the corresponding eigenfunctions of an associated pair of regularized operators, similar to the Euclidean case. The known weak convergence of the sample covariance operator, coupled with a delta-method for analytic functions of covariance operators, yields the weak convergence of the pair of associated operators. From the latter we…
The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to recent state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modific…
Cauchy Aggregation of Ridge-Regularized Hotelling Tests for High-Dimensional Change-Point Detection
School of Mathematical Science, Tiangong University（天津工业大学数学科学学院）；Department of Mathematics, Hong Kong Baptist University（香港浸会大学数学系）；School of Statistics and Data Science, Nankai University（南开大学统计与数据科学学院）
In this paper we propose a new regression interpretation of the Cholesky factor of the covariance matrix, as opposed to the well known regression interpretation of the Cholesky factor of the inverse covariance, which leads to a new class of regularized covariance estimators suitable for high-dimensional problems. Regularizing the Cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator. In particular, one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estim…
In this paper, we propose an abstract procedure for debiasing constrained or regularized potentially high-dimensional linear models. It is elementary to show that the proposed procedure can produce $\frac{1}{\sqrt{n}}$-confidence intervals for individual coordinates (or even bounded contrasts) in models with unknown covariance, provided that the covariance has bounded spectrum. While the proof of the statistical guarantees of our procedure is simple, its implementation requires more care due to the complexity of the optimization programs we need to solve. We spend the bulk of this paper giving…
Robust estimators of large covariance matrices are considered, comprising regularized (linear shrinkage) modifications of Maronna's classical M-estimators. These estimators provide robustness to outliers, while simultaneously being well-defined when the number of samples does not exceed the number of variables. By applying tools from random matrix theory, we characterize the asymptotic performance of such estimators when the numbers of samples and variables grow large together. In particular, our results show that, when outliers are absent, many estimators of the regularized-Maronna type share…
This paper considers estimating a covariance matrix of $p$ variables from $n$ observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance. We show that these estimates are consistent in the operator norm as long as $(\log p)/n\to0$, and obtain explicit rates. The results are uniform over some fairly natural well-conditioned families of covariance matrices. We also introduce an analogue of the Gaussian white noise model and show that if the population covariance is embeddable in that model and well-conditioned, then th…
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）
We consider the problem of large-scale inference on the row or column variables of data in the form of a matrix. Often this data is transposable, meaning that both the row variables and column variables are of potential interest. An example of this scenario is detecting significant genes in microarrays when the samples or arrays may be dependent due to underlying relationships. We study the effect of both row and column correlations on commonly used test-statistics, null distributions, and multiple testing procedures, by explicitly modeling the covariances with the matrix-variate normal distri…
Spectrally-Corrected and Regularized QDA Classifier for Spiked Covariance Model
School of Data Sciences, Zhejiang University of Finance and Economics（浙江财经大学数据科学学院）；School of Mathematics and Statistics, Northeast Normal University（东北师范大学数学与统计学院）；Northeast Normal University（东北师范大学）
From Cross-Validation to SURE: Asymptotic Risk of Tuned Regularized Estimators
Department of Economics, University of Pennsylvania（宾夕法尼亚大学经济学系）；Department of Economics, University of Oxford（牛津大学经济学系）；Department of Electrical Engineering and Computer Science, MIT（麻省理工学院电气工程与计算机科学系）
We provide a unified analysis of the predictive risk of ridge regression and regularized discriminant analysis in a dense random effects model. We work in a high-dimensional asymptotic regime where $p, n \to \infty$ and $p/n \to γ\in (0, \, \infty)$, and allow for arbitrary covariance among the features. For both methods, we provide an explicit and efficiently computable expression for the limiting predictive risk, which depends only on the spectrum of the feature-covariance matrix, the signal strength, and the aspect ratio $γ$. Especially in the case of regularized discriminant analysis, we f…
Shrinkage estimators of covariance are an important tool in modern applied and theoretical statistics. They play a key role in regularized estimation problems, such as ridge regression (aka Tykhonov regularization), regularized discriminant analysis and a variety of optimization problems. In this paper, we bring to bear the tools of random matrix theory to understand their behavior, and in particular, that of quadratic forms involving inverses of those estimators, which are important in practice. We use very mild assumptions compared to the usual assumptions made in random matrix theory, requi…
We study the problem of estimating an unknown vector $θ$ from an observation $X$ drawn according to the normal distribution with mean $θ$ and identity covariance matrix under the knowledge that $θ$ belongs to a known closed convex set $Θ$. In this general setting, Chatterjee (2014) proved that the natural constrained least squares estimator is "approximately admissible" for every $Θ$. We extend this result by proving that the same property holds for all convex penalized estimators as well. Moreover, we simplify and shorten the original proof considerably. We also provide explicit upper and low…
Estimating a high-dimensional sparse covariance matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards bridging this gap, in this work we consider estimating a sparse shape matrix from $n$ samples following a possibly heavy tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We derive bounds on the difference in spectral norm between our estimators and the shape matrix in the joint limit as th…
We propose a novel estimation approach for the covariance matrix based on the $l_1$-regularized approximate factor model. Our sparse approximate factor (SAF) covariance estimator allows for the existence of weak factors and hence relaxes the pervasiveness assumption generally adopted for the standard approximate factor model. We prove consistency of the covariance matrix estimator under the Frobenius norm as well as the consistency of the factor loadings and the factors. Our Monte Carlo simulations reveal that the SAF covariance estimator has superior properties in finite samples for low and h…
This paper considers the banding estimator proposed in Bickel and Levina (2008) for estimation of large covariance matrices. We prove that the banding estimator achieves rate-optimality under the operator norm, for a class of approximately banded covariance matrices, improving the existing results in Bickel and Levina (2008). In addition, we propose a Stein's unbiased risk estimate (Sure)-type approach for selecting the bandwidth for the banding estimator. Simulations indicate that the Sure-tuned banding estimator outperforms competing estimators.
The optimal transport map between the standard Gaussian measure and an $α$-strongly log-concave probability measure is $α^{-1/2}$-Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp-Lieb and Cramér-Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli's original result. We also extend Caffarelli's theorem to the setting …
Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity between these predictors or small sample sizes compared to the dimension, the inversion is not possible and a regularization technique has to be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it is shown that SIR axes can be interpreted as solutions of an inverse regression problem. In this paper, a Gaussian prior distribution is introduc…
We introduce a new second order stochastic algorithm to estimate the entropically regularized optimal transport cost between two probability measures. The source measure can be arbitrary chosen, either absolutely continuous or discrete, while the target measure is assumed to be discrete. To solve the semi-dual formulation of such a regularized and semi-discrete optimal transportation problem, we propose to consider a stochastic Gauss-Newton algorithm that uses a sequence of data sampled from the source measure. This algorithm is shown to be adaptive to the geometry of the underlying convex opt…
We give oracle inequalities on procedures which combines quantization and variable selection via a weighted Lasso $k$-means type algorithm. The results are derived for a general family of weights, which can be tuned to size the influence of the variables in different ways. Moreover, these theoretical guarantees are proved to adapt the corresponding sparsity of the optimal codebooks, if appropriate. Even if there is no sparsity assumption on the optimal codebooks, our procedure is proved to be close to a sparse approximation of the optimal codebooks, as has been done for the Generalized Linear …
Graphical model selection is a seemingly impossible task when many pairs of variables are never jointly observed; this requires inference of conditional dependencies with no observations of corresponding marginal dependencies. This under-explored statistical problem arises in neuroimaging, for example, when different partially overlapping subsets of neurons are recorded in non-simultaneous sessions. We call this statistical challenge the "Graph Quilting" problem. We study this problem in the context of sparse inverse covariance learning, and focus on Gaussian graphical models where we show tha…
This paper examines the usefulness of high frequency data in estimating the covariance matrix for portfolio choice when the portfolio size is large. A computationally convenient nonlinear shrinkage estimator for the integrated covariance (ICV) matrix of financial assets is developed in two steps. The eigenvectors of the ICV are first constructed from a designed time variation adjusted realized covariance matrix of noise-free log-returns of relatively low frequency data. Then the regularized eigenvalues of the ICV are estimated by quasi-maximum likelihood based on high frequency data. The estim…
We consider the problem of computing a positive definite $p \times p$ inverse covariance matrix aka precision matrix $θ=(θ_{ij})$ which optimizes a regularized Gaussian maximum likelihood problem, with the elastic-net regularizer $\sum_{i,j=1}^{p} λ(α|θ_{ij}| + \frac{1}{2}(1- α) θ_{ij}^2),$ with regularization parameters $α\in [0,1]$ and $λ>0$. The associated convex semidefinite optimization problem is notoriously difficult to scale to large problems and has demanded significant attention over the past several years. We propose a new algorithmic framework based on stochastic proximal optimizat…
