Estimating Sparse Precision Matrix: Optimal Rates of Convergence and Adaptive Estimation

By Tommaso Cai, Weidong Liu, Harrison H. Zhou. 2012

220 citations recorded by the source corpus.

Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained l1 minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces. The estimator, called ACLIME, is easy to implement and performs well numerically. A major step in establishing the minimax rate of convergence is the derivation of a rate-sharp lower bound. A “two-directional” lower bound technique is applied to obtain the minimax lower bound. The upper and lower bounds together yield the optimal rates of convergence for sparse precision matrix estimation and show that the ACLIME estimator is adaptively minimax rate optimal for a collection of parameter spaces and a range of matrix norm losses simultaneously.

Subfields: Computational Mechanics, Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ba3bbed50c5cb80f1692785852c86d6704f25207

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Citations in the corpus

Robust regression via mutivariate regression depth (2017, background): “The first term R(0) has order s(2) log p n , which was proved in [5].”
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, methodology): “Specifically, suppose the random vector X = (X1, . . . , Xp) ′ consists of the first p variables of a fixed zero mean stationary process {Xi} as p → ∞.”
Asymptotic normality and optimalities in estimation of large Gaussian graphical models (2013, unknown): “Statist. 36 199-227. Bickel, P. J. and Levina, E. (2008b). Covariance regularization by thresholding.”
Uniform Inference for High-dimensional Quantile Regression: Linear Functionals and Regression Rank Scores (2017, background): “For example, assumption (X) was considered in the literature on inverse covariance matrix estimation (Cai et al., 2016) and in a slightly weaker form for inference of univariate parameters (van de Geer et al.”
Large Scale Distributed Sparse Precision Estimation (2013, methodology): “Spurred by these advances in the statistical theory of precision matrix estimation, there has been considerable recent work on developing computationally efficient optimization methods for solving the corresponding statistical estimation problems: see [1, 8, 14, 21, 13], and references therein.”
Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures (2017, background): “See Cai et al. (2016) for a survey of the recent results on optimal and adaptive estimation, and for a discussion of some key technical tools used in the theoretical analyses. www.annualreviews.org • Hypothesis Testing for Covariance Structures 3 Parallel to estimation, there have been significant…”
Structure Learning in Graphical Modeling (2016, background): “Minimax optimality properties can be proven for an adaptive version of the estimator (Cai et al., 2016).”