In this paper we consider estimation of sparse covariance matrices and propose a thresholding procedure which is adaptive to the variability of individual entries. The estimators are fully data driven and enjoy excellent performance both theoretically and numerically. It is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm. In contrast, the commonly used universal thresholding estimators are shown to be sub-optimal over the same parameter spaces. Support recovery is also discussed. The adaptive thresholding estimators are easy to implement. Numerical performance of the estimators is studied using both simulated and real data. Simulation results show that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators. The method is also illustrated in an analysis on a dataset from a small round blue-cell tumors microarray experiment. A supplement to this paper which contains additional technical proofs is available online.
High-dimensional compositional data arise naturally in many applications such as metagenomic data analysis. The observed data lie in a high-dimensional simplex, and conventional statistical methods often fail to produce sensible results due to the unit-sum constraint. In this article, we address the problem of covariance estimation for high-dimensional compositional data, and introduce a composition-adjusted thresholding (COAT) method under the assumption that the basis covariance matrix is sparse. Our method is based on a decomposition relating the compositional covariance to the basis covari…
In high-dimensions, many variable selection methods, such as the lasso, are often limited by excessive variability and rank deficiency of the sample covariance matrix. Covariance sparsity is a natural phenomenon in high-dimensional applications, such as microarray analysis, image processing, etc., in which a large number of predictors are independent or weakly correlated. In this paper, we propose the covariance-thresholded lasso, a new class of regression methods that can utilize covariance sparsity to improve variable selection. We establish theoretical results, under the random design setti…
We consider testing the equality of two high-dimensional covariance matrices by carrying out a multi-level thresholding procedure, which is designed to detect sparse and faint differences between the covariances. A novel U-statistic composition is developed to establish the asymptotic distribution of the thresholding statistics in conjunction with the matrix blocking and the coupling techniques. We propose a multi-thresholding test that is shown to be powerful in detecting sparse and weak differences between two covariance matrices. The test is shown to have attractive detection boundary and t…
Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this paper, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions p is comparable to, or even larger than the sample size n. Aided by the Hilbert--Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding estimator by incorporating the variance effects o…
We consider the problem of clustering functional data while jointly selecting the most relevant features for classification. This problem has never been tackled before in the functional data context, and it requires a proper definition of the concept of sparsity for functional data. Functional sparse clustering is here analytically defined as a variational problem with a hard thresholding constraint ensuring the sparsity of the solution. First, a unique solution to sparse clustering with hard thresholding in finite dimensions is proved to exist. Then, the infinite dimensional generalization is…
We consider a class of Cox models with time-dependent effects that may be zero over certain unknown time regions or, in short, sparse time-varying effects. The model is particularly useful for biomedical studies as it conveniently depicts the gradual evolution of effects of risk factors on survival. Statistically, estimating and drawing inference on infinite dimensional functional parameters with sparsity (e.g., time-varying effects with zero-effect time intervals) present enormous challenges. To address them, we propose a new soft-thresholding operator for modeling sparse, piecewise smooth an…
Determining the number of common factors is an important and practical topic in high dimensional factor models. The existing literatures are mainly based on the eigenvalues of the covariance matrix. Due to the incomparability of the eigenvalues of the covariance matrix caused by heterogeneous scales of observed variables, it is very difficult to give an accurate relationship between these eigenvalues and the number of common factors. To overcome this limitation, we appeal to the correlation matrix and show surprisingly that the number of eigenvalues greater than $1$ of population correlation m…
Soft-thresholding is a sparse modeling method that is typically applied to wavelet denoising in statistical signal processing and analysis. It has a single parameter that controls a threshold level on wavelet coefficients and, simultaneously, amount of shrinkage for coefficients of un-removed components. This parametrization is possible to cause excess shrinkage, thus, estimation bias at a sparse representation; i.e. there is a dilemma between sparsity and prediction accuracy. To relax this problem, we considered to introduce positive scaling on soft-thresholding estimator, by which threshold …
We consider testing for two-sample means of high dimensional populations by thresholding. Two tests are investigated, which are designed for better power performance when the two population mean vectors differ only in sparsely populated coordinates. The first test is constructed by carrying out thresholding to remove the non-signal bearing dimensions. The second test combines data transformation via the precision matrix with the thresholding. The benefits of the thresholding and the data transformations are showed by a reduced variance of the test thresholding statistics, the improved power an…
Microbial communities analysis is drawing growing attention due to the rapid development of high-throughput sequencing techniques nowadays. The observed data has the following typical characteristics: it is high-dimensional, compositional (lying in a simplex) and even would be leptokurtic and highly skewed due to the existence of overly abundant taxa, which makes the conventional correlation analysis infeasible to study the co-occurrence and co-exclusion relationship between microbial taxa. In this article, we address the challenges of covariance estimation for this kind of data. Assuming the …
Principal component analysis (PCA) is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix. However, it behaves poorly when the number of features p is comparable to, or even much larger than, the sample size n. In this paper, we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse. Under a spiked covariance model, we find that the new approach recovers the principal subspace and leading eigenvectors consistently, an…
Adaptive thresholding methods have proved to yield high SNRs and fast convergence in finding the solution to the Compressed Sensing (CS) problems. Recently, it was observed that the robustness of a class of iterative sparse recovery algorithms such as Iterative Method with Adaptive Thresholding (IMAT) has outperformed the well-known LASSO algorithm in terms of reconstruction quality, convergence speed, and the sensitivity to the noise. In this paper, we introduce a new method towards solving the CS problem. The logic of this method is based on iterative projections of the thresholded signal on…
Smoothing of noisy sample covariances is an important component in functional data analysis. We propose a novel covariance smoothing method based on penalized splines and associated software. The proposed method is a bivariate spline smoother that is designed for covariance smoothing and can be used for sparse functional or longitudinal data. We propose a fast algorithm for covariance smoothing using leave-one-subject-out cross validation. Our simulations show that the proposed method compares favorably against several commonly used methods. The method is applied to a study of child growth led…
We present a graph-based technique for estimating sparse covariance matrices and their inverses from high-dimensional data. The method is based on learning a directed acyclic graph (DAG) and estimating parameters of a multivariate Gaussian distribution based on a DAG. For inferring the underlying DAG we use the PC-algorithm and for estimating the DAG-based covariance matrix and its inverse, we use a Cholesky decomposition approach which provides a positive (semi-)definite sparse estimate. We present a consistency result in the high-dimensional framework and we compare our method with the Glass…
We propose an l1-regularized likelihood method for estimating the inverse covariance matrix in the high-dimensional multivariate normal model in presence of missing data. Our method is based on the assumption that the data are missing at random (MAR) which entails also the completely missing at random case. The implementation of the method is non-trivial as the observed negative log-likelihood generally is a complicated and non-convex function. We propose an efficient EM algorithm for optimization with provable numerical convergence properties. Furthermore, we extend the methodology to handle …
Sparse Canonical Correlation Analysis (CCA) has received considerable attention in high-dimensional data analysis to study the relationship between two sets of random variables. However, there has been remarkably little theoretical statistical foundation on sparse CCA in high-dimensional settings despite active methodological and applied research activities. In this paper, we introduce an elementary sufficient and necessary characterization such that the solution of CCA is indeed sparse, propose a computationally efficient procedure, called CAPIT, to estimate the canonical directions, and show…
We consider the problem of estimating sparse graphs by a lasso penalty applied to the inverse covariance matrix. Using a coordinate descent procedure for the lasso, we develop a simple algorithm that is remarkably fast: in the worst cases, it solves a 1000 node problem (~500,000 parameters) in about a minute, and is 50 to 2000 times faster than competing methods. It also provides a conceptual link between the exact problem and the approximation suggested by Meinhausen and Buhlmann (2006). We illustrate the method on some cell-signaling data from proteomics.
We propose a general framework for nonasymptotic covariance matrix estimation making use of concentration inequality-based confidence sets. We specify this framework for the estimation of large sparse covariance matrices through incorporation of past thresholding estimators with key emphasis on support recovery. This technique goes beyond past results for thresholding estimators by allowing for a wide range of distributional assumptions beyond merely sub-Gaussian tails. This methodology can furthermore be adapted to a wide range of other estimators and settings. The usage of nonasymptotic dime…
The thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices, but it often has negative eigenvalues when used in real data analysis. To simultaneously achieve sparsity and positive definiteness, we develop a positive definite $\ell_1$-penalized covariance estimator for estimating sparse large covariance matrices. An efficient alternating direction method is derived to solve the challenging optimization problem and its convergence properties are established. Under weak regularity conditions, non-asymptotic statistical theory is also establ…
We consider a high-dimensional linear regression problem. Unlike many papers on the topic, we do not require sparsity of the regression coefficients; instead, our main structural assumption is a decay of eigenvalues of the covariance matrix of the data. We propose a new family of estimators, called the canonical thresholding estimators, which pick largest regression coefficients in the canonical form. The estimators admit an explicit form and can be linked to LASSO and Principal Component Regression (PCR). A theoretical analysis for both fixed design and random design settings is provided. Obt…
Covariance estimation is essential yet underdeveloped for analyzing multivariate functional data. We propose a fast covariance estimation method for multivariate sparse functional data using bivariate penalized splines. The tensor-product B-spline formulation of the proposed method enables a simple spectral decomposition of the associated covariance operator and explicit expressions of the resulting eigenfunctions as linear combinations of B-spline bases, thereby dramatically facilitating subsequent principal component analysis. We derive a fast algorithm for selecting the smoothing parameters…
We derive a formula for optimal hard thresholding of the singular value decomposition in the presence of correlated additive noise; although it nominally involves unobservables, we show how to apply it even where the noise covariance structure is not a-priori known or is not independently estimable. The proposed method, which we call ScreeNOT, is a mathematically solid alternative to Cattell's ever-popular but vague Scree Plot heuristic from 1966. ScreeNOT has a surprising oracle property: it typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on e…
The University of Electro-Communications（电气通信大学）；National Police Agency Info-Communications Bureau（警察厅信息通信局）
In this paper, we propose a Bayesian Graphical LASSO for correlated countable data and apply it to spatial crime data. In the proposed model, we assume a Gaussian Graphical Model for the latent variables which dominate the potential risks of crimes. To evaluate the proposed model, we determine optimal hyperparameters which represent samples better. We apply the proposed model for estimation of the sparse inverse covariance of the latent variable and evaluate the partial correlation coefficients. Finally, we illustrate the results on crime spots data and consider the estimated latent variables …
Statistical inference for sparse covariance matrices is crucial to reveal dependence structure of large multivariate data sets, but lacks scalable and theoretically supported Bayesian methods. In this paper, we propose beta-mixture shrinkage prior, computationally more efficient than the spike and slab prior, for sparse covariance matrices and establish its minimax optimality in high-dimensional settings. The proposed prior consists of beta-mixture shrinkage and gamma priors for off-diagonal and diagonal entries, respectively. To ensure positive definiteness of the resulting covariance matrix,…
This paper introduces a new data-driven methodology for estimating sparse covariance matrices of the random coefficients in logit mixture models. Researchers typically specify covariance matrices in logit mixture models under one of two extreme assumptions: either an unrestricted full covariance matrix (allowing correlations between all random coefficients), or a restricted diagonal matrix (allowing no correlations at all). Our objective is to find optimal subsets of correlated coefficients for which we estimate covariances. We propose a new estimator, called MISC, that uses a mixed-integer op…
A new framework is developed to intrinsically analyze sparsely observed Riemannian functional data. It features four innovative components: a frame-independent covariance function, a smooth vector bundle termed covariance vector bundle, a parallel transport and a smooth bundle metric on the covariance vector bundle. The introduced intrinsic covariance function links estimation of covariance structure to smoothing problems that involve raw covariance observations derived from sparsely observed Riemannian functional data, while the covariance vector bundle provides a rigorous mathematical founda…
We propose methodology for statistical inference for low-dimensional parameters of sparse precision matrices in a high-dimensional setting. Our method leads to a non-sparse estimator of the precision matrix whose entries have a Gaussian limiting distribution. Asymptotic properties of the novel estimator are analyzed for the case of sub-Gaussian observations under a sparsity assumption on the entries of the true precision matrix and regularity conditions. Thresholding the de-sparsified estimator gives guarantees for edge selection in the associated graphical model. Performance of the proposed m…
We study the estimation of the covariance matrix $Σ$ of a $p$-dimensional normal random vector based on $n$ independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of $Σ$. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are …
We consider Bayesian inference of sparse covariance matrices and propose a post-processed posterior. This method consists of two steps. In the first step, posterior samples are obtained from the conjugate inverse-Wishart posterior without considering the sparse structural assumption. The posterior samples are transformed in the second step to satisfy the sparse structural assumption through the hard-thresholding function. This non-traditional Bayesian procedure is justified by showing that the post-processed posterior attains the optimal minimax rates. We also investigate the application of th…
We offer a method to estimate a covariance matrix in the special case that \textit{both} the covariance matrix and the precision matrix are sparse --- a constraint we call double sparsity. The estimation method is maximum likelihood, subject to the double sparsity constraint. In our method, only a particular class of sparsity pattern is allowed: both the matrix and its inverse must be subordinate to the same chordal graph. Compared to a naive enforcement of double sparsity, our chordal graph approach exploits a special algebraic local inverse formula. This local inverse property makes computat…
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…
