Estimating structured high-dimensional covariance and precision matrices: optimal rates and adaptive estimation

DOI10.1214/15-EJS1081zbMath1331.62272MaRDI QIDQ5965313

Zhao Ren, T. Tony Cai, Harrison H. Zhou

Publication date: 3 March 2016

Published in: Electronic Journal of Statistics (Search for Journal in Brave)

Full work available at URL: https://projecteuclid.org/euclid.ejs/1455715952

zbMATH Keywords

hypothesis testingcovariance matrixFrobenius normminimax lower boundthresholdingadaptive estimationoperator normoptimal rate of convergencefactor modelspectral normbandingblock thresholdingGaussian graphical modelprecision matrixSchatten normtapering

Mathematics Subject Classification ID

Asymptotic properties of parametric estimators (62F12) Factor analysis and principal components; correspondence analysis (62H25) Estimation in multivariate analysis (62H12) Nonparametric statistical resampling methods (62G09)

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Uses Software

• glasso
• FASTCLIME
• QUIC
• PMA

Cites Work

• On asymptotically optimal confidence regions and tests for high-dimensional models
• Regularized estimation in sparse high-dimensional time series models
• Sparse inverse covariance estimation with the graphical lasso
• Latent variable graphical model selection via convex optimization
• Asymptotic normality and optimalities in estimation of large Gaussian graphical models
• Gaussian graphical model estimation with false discovery rate control
• Estimating sparse precision matrix: optimal rates of convergence and adaptive estimation
• High dimensional covariance matrix estimation using a factor model
• Statistical and computational trade-offs in estimation of sparse principal components
• Sparse principal component analysis and iterative thresholding
• Minimax bounds for sparse PCA with noisy high-dimensional data
• Asymptotic power of sphericity tests for high-dimensional data
• Optimal detection of sparse principal components in high dimension
• Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors
• Accuracy of the Tracy-Widom limits for the extreme eigenvalues in white Wishart matrices
• Likelihood ratio tests for covariance matrices of high-dimensional normal distributions
• On Jiang's asymptotic distribution of the largest entry of a sample correlation matrix
• Noisy matrix decomposition via convex relaxation: optimal rates in high dimensions
• High-dimensional covariance matrix estimation in approximate factor models
• Covariance matrix estimation for stationary time series
• Tracy-Widom distribution for the largest eigenvalue of real sample covariance matrices with general population
• Estimation of high-dimensional low-rank matrices
• Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices
• Anisotropic local laws for random matrices
• Test for bandedness of high-dimensional covariance matrices and bandwidth estimation
• Optimal rates of convergence for sparse covariance matrix estimation
• The application of parametric multichannel spectral estimates in the study of electrical brain activity
• On the estimation of quadratic functionals
• High-dimensional analysis of semidefinite relaxations for sparse principal components
• Asymptotic equivalence of spectral density estimation and Gaussian white noise
• Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter vari\-ants
• Sparse principal component analysis via regularized low rank matrix approximation
• Optimal rates of convergence for covariance matrix estimation
• Covariance regularization by thresholding
• Operator norm consistent estimation of large-dimensional sparse covariance matrices
• Finite sample approximation results for principal component analysis: A matrix perturbation approach
• Universality results for the largest eigenvalues of some sample covariance matrix ensembles
• Corrections to LRT on large-dimensional covariance matrix by RMT
• Sparsistency and rates of convergence in large covariance matrix estimation
• Geometrizing rates of convergence. II
• The strong limits of random matrix spectra for sample matrices of independent elements
• A Cramér type large deviation result for Student's \(t\)-statistic
• A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices
• Information-theoretic determination of minimax rates of convergence
• Sparse CCA: adaptive estimation and computational barriers
• Some theory for Fisher's linear discriminant function, `naive Bayes', and some alternatives when there are many more variables than observations
• On the distribution of the largest eigenvalue in principal components analysis
• Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size
• The asymptotic distributions of the largest entries of sample correlation matrices.
• Adaptive covariance matrix estimation through block thresholding
• Consistency of sparse PCA in high dimension, low sample size contexts
• Sparse permutation invariant covariance estimation
• On the conditions used to prove oracle results for the Lasso
• High-dimensional covariance estimation by minimizing \(\ell _{1}\)-penalized log-determinant divergence
• Optimal rates of convergence for estimating Toeplitz covariance matrices
• Shape fluctuations and random matrices
• Optimal estimation and rank detection for sparse spiked covariance matrices
• Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing
• Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions
• Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices
• The asymptotic distribution and Berry-Esseen bound of a new test for independence in high dimension with an application to stochastic optimization
• Simultaneous analysis of Lasso and Dantzig selector
• Optimal hypothesis testing for high dimensional covariance matrices
• Minimax sparse principal subspace estimation in high dimensions
• Covariance and precision matrix estimation for high-dimensional time series
• Sparse PCA: optimal rates and adaptive estimation
• Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices
• Pathwise coordinate optimization
• Regularized estimation of large covariance matrices
• High-dimensional graphs and variable selection with the Lasso
• Convergence of estimates under dimensionality restrictions
• On some test criteria for covariance matrix
• A note on testing the covariance matrix for large dimension
• Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
• Nonparametric estimation of large covariance matrices of longitudinal data
• Sparse Matrix Inversion with Scaled Lasso
• Hypothesis Testing in High-Dimensional Regression Under the Gaussian Random Design Model: Asymptotic Theory
• Reconstruction From Anisotropic Random Measurements
• Robust principal component analysis?
• A Constrainedℓ₁Minimization Approach to Sparse Precision Matrix Estimation
• Adaptive Thresholding for Sparse Covariance Matrix Estimation
• Banded and tapered estimates for autocovariance matrices and the linear process bootstrap
• Scaled sparse linear regression
• A direct approach to sparse discriminant analysis in ultra-high dimensions
• A Direct Estimation Approach to Sparse Linear Discriminant Analysis
• On Consistent Estimates of the Spectrum of a Stationary Time Series
• Model selection and estimation in the Gaussian graphical model
• Asymptotic distribution of the largest off-diagonal entry of correlation matrices
• First-Order Methods for Sparse Covariance Selection
• Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets
• Determinant Maximization with Linear Matrix Inequality Constraints
• Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings
• On Consistency and Sparsity for Principal Components Analysis in High Dimensions
• Tests for High-Dimensional Covariance Matrices
• Generalized Thresholding of Large Covariance Matrices
• Nonlinear system theory: Another look at dependence
• Covariance matrix selection and estimation via penalised normal likelihood
• The Rotation of Eigenvectors by a Perturbation. III
• Robust Portfolio Selection Problems
• A Direct Formulation for Sparse PCA Using Semidefinite Programming
• Introduction to nonparametric estimation
• Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix
• A general theory of concave regularization for high-dimensional sparse estimation problems
• Discussion: Latent variable graphical model selection via convex optimization
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