Generalized Thresholding of Large Covariance Matrices

By Adam J. Rothman, Elizaveta Levina, Ji Zhu. 2009

536 citations recorded by the source corpus.

Subfields: Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ea421ccff0d7a0a52369ca24a959911ba741cc0b

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In Computational Mechanics

In Statistics and Probability

Citations in the corpus

An Overview on the Estimation of Large Covariance and Precision Matrices (2015, methodology): “To estimate Σ, Liu et al. (2014b) introduce a regularized rank-based estimation method named EC2 (Estimation of Covariance with Eigenvalue Constraints), which can be viewed as an extension of the generalized thresholding operator (Rothman et al., 2009).”
Efficient Estimation of Approximate Factor Models (2012, methodology): “Define a general thresh- olding function sij(z) : R → R as in Rothman et al. (2009) and Cai and Liu (2011) with an entry-dependent threshold τij such that: (i) sij(z) = 0 if |z| < τij ; (ii) |sij(z)− z| ≤ τij .”
Adaptive Thresholding for Sparse Covariance Matrix Estimation (2011, unknown): “For sparse covariance matrix estimation, the following uniformity class of sparse matrices was considered by Bickel and Levina (2008) and Rothman, Levina, and Zhu (2009): U q := U q (s 0 ( p)) = : 0, max i σ ii ≤ K, max i p j=1 |σ ij | q ≤ s 0 (p) for some 0 ≤ q 1, where 0 means that is positive…”
High‐dimensional covariance matrix estimation (2019, methodology): “This estimator is also proved to be asymptotically as efficient as Σ̂Ideal in estimating Σ with respect to the Frobenius loss when p/n! c>0, while its inverse is asymptotically as efficient as Σ̂ Ideal in estimating Σ −1 with respect to the inverse Stein's loss under p/n! c>0. Lam (2016) also proved”
Positive Definite $\ell_1$ Penalized Estimation of Large Covariance Matrices (2012, methodology): “For all three regularized estimators, the thresholding parameter was chosen by 5-fold cross-validation (Bickel and Levina, 2008b; Rothman et al., 2009; Cai and Liu, 2011).”
Informative Estimation and Selection of Correlation Structure for Longitudinal Data (2012, methodology): “Other estimation approaches for high-dimensional covariance matrices include the nested least absolute shrinkage and selection operator (LASSO) approach (Levina, Rothman, and Zhu 2008) and thresholding approaches (Bickel and Levina 2008; Rothman, Levina, and Zhu 2009).”
Nonparametric eigenvalue-regularized precision or covariance matrix estimator (2016, methodology): “Our method avoids the need to first estimate any unknowns from a factor model, and directly gives the covariance or precision matrix estimator, which can be useful when factor analysis is not the ultimate goal.”
Covariance Estimation: The GLM and Regularization Perspectives (2011, background): “From (17) it is evident that, the orthogonalization process reduces the task of modeling a covariance matrix to that of a sequence of p varying-coefficient and varyingorder regression models so that one can bring the whole regression analysis machinery to the service of the unintuitive task of model”
Using Principal Component Analysis to Estimate a High Dimensional Factor Model with High-Frequency Data (2017, methodology): “Since Γ is assumed sparse, we can enforce sparsity through, e.g., soft-, hard-, or adaptive thresholding; see, e.g., Rothman et al. (2009) for a discussion of thresholding techniques.”
Large Covariance Estimation by Thresholding Principal Orthogonal Complements (2011, unknown): “This section compares POET with direct thresholding on the sample covariance matrix without taking out common factors (Rothman et al. 2009, Cai and Liu 2011.”