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

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

doi:10.1214/15-ejs1081

Cited by 157 publications

(146 citation statements)

References 124 publications

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“…In practice, Σ has to be estimated. We may apply an existing method, such as the banding and thresholding technique, to estimate a high-dimensional sparse covariance matrix (Bickel and Levina, 2008;Cai and Liu, 2011); see Cai et al (2016) for an excellent review. Under the α mixing assumption C2, σ ij is close to zero when |i − j| is large and thus we may apply the banding approach of Bickel and Levina (2008) …”

Section: C9 the Conditionally α-Mixingmentioning

confidence: 99%

An Adaptive Test on High-dimensional Parameters in Generalized Linear Models

Pan³

2019

STAT SINICA

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Significance testing for high-dimensional generalized linear models (GLMs) has been increasingly needed in various applications, however, existing methods are mainly based on a sum of squares of the score vector and only powerful under certain alternative hypotheses. In practice, depending on whether the true association pattern under an alternative hypothesis is sparse or dense or between, the existing tests may or may not be powerful. In this paper, we propose an adaptive test on a high-dimensional parameter of a GLM (in the presence of a low-dimensional nuisance parameter), which can maintain high power across a wide range of scenarios. To evaluate its p-value, its asymptotic null distribution is derived. We conduct simulations to demonstrate the superior performance of the proposed test. In addition, we apply it and other existing tests to an Alzheimer's Disease Neuroimaging Initiative (ADNI) * Correspondence: wuxx0845@umn.edu (C.W.), weip@biostat.umn.edu (W.P.) † Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at:

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Section: C9 the Conditionally α-Mixingmentioning

confidence: 99%

An Adaptive Test on High-dimensional Parameters in Generalized Linear Models

Pan³

2019

STAT SINICA

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“…There is a large body of work on high dimensional covariance and precision matrix estimation: see for example the recent review paper of Cai et al (2016) and references therein. Much of the work on the specific setting with latent confounding has focused on estimation of the precision matrix Ω, which is assumed to be sparse.…”

Section: Related Workmentioning

confidence: 99%

Right Singular Vector Projection Graphs: Fast High Dimensional Covariance Matrix Estimation under Latent Confounding

Shah

Frot

Thanei

et al. 2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary We consider the problem of estimating a high dimensional p×p covariance matrix Σ, given n observations of confounded data with covariance Σ+ΓnormalΓsans-serifT, where Γ is an unknown p×q matrix of latent factor loadings. We propose a simple and scalable estimator based on the projection onto the right singular vectors of the observed data matrix, which we call right singular vector projection (RSVP). Our theoretical analysis of this method reveals that, in contrast with approaches based on the removal of principal components, RSVP can cope well with settings where the smallest eigenvalue of normalΓsans-serifTΓ is relatively close to the largest eigenvalue of Σ, as well as when the eigenvalues of normalΓsans-serifTΓ are diverging fast. RSVP does not require knowledge or estimation of the number of latent factors q, but it recovers Σ only up to an unknown positive scale factor. We argue that this suffices in many applications, e.g. if an estimate of the correlation matrix is desired. We also show that, by using subsampling, we can further improve the performance of the method. We demonstrate the favourable performance of RSVP through simulation experiments and an analysis of gene expression data sets collated by the GTEX consortium.

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“…Most methods focus on classes of matrices with some sparsity constraint, either on the covariance matrix itself or on its inverse, the precision matrix. See [10] for a recent survey. Among these matrix classes, the one that fits more naturally our local patch models is the spiked covariance, which expresses the covariance matrix as a low-rank signal component plus noise (AGWN).…”

Section: Empirical Wiener Filters For Denoisingmentioning

confidence: 99%

Video Denoising via Empirical Bayesian Estimation of Space-Time Patches

Arias

Morel

2017

J Math Imaging Vis

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In this paper we present a new patch-based empirical Bayesian video denoising algorithm. The method builds a Bayesian model for each group of similar space-time patches. These patches are not motion compensated, and therefore avoid the risk of inaccuracies caused by motion estimation errors. The high dimensionality of spatio-temporal patches together with a limited number of available samples, poses challenges when estimating the statistics needed for an empirical Bayesian method. We therefore assume that groups of similar patches have a low intrinsic dimensionality, leading to a spiked covariance model. Based on theoretical results about the estimation of spiked covariance matrices, we propose estimators of the eigenvalues of the a priori covariance in high dimensional spaces as simple corrections of the eigenvalues of the sample covariance matrix. We demonstrate empirically that these estimators lead to better empirical Wiener filters. A comparison on classic benchmark videos demonstrates improved visual quality and an increased PSNR with respect to state-of-the-art video denoising methods.

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

Cited by 157 publications

References 124 publications

An Adaptive Test on High-dimensional Parameters in Generalized Linear Models

An Adaptive Test on High-dimensional Parameters in Generalized Linear Models

Right Singular Vector Projection Graphs: Fast High Dimensional Covariance Matrix Estimation under Latent Confounding

Video Denoising via Empirical Bayesian Estimation of Space-Time Patches

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