Large-Dimensional Behavior of Regularized Maronna's M-Estimators of Covariance Matrices

Auguin, Nicolas; Morales-Jiménez, David; McKay, Matthew R.; Couillet, Romain

doi:10.1109/tsp.2018.2831629

Cited by 7 publications

(5 citation statements)

References 35 publications

(113 reference statements)

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“…In [26], this analysis was extended to X having an elliptical distribution with a general scatter matrix. A similar asymptotic analysis for MRE appeared in [6]. Two variants of TRE were studied in [24], assuming X has an elliptical distribution.…”

Section: Introductionmentioning

confidence: 92%

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Tyler's and Maronna's M-estimators: Non-Asymptotic Concentration Results

Romanov¹,

Kur²,

Nadler³

2022

Preprint

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Tyler's and Maronna's M-estimators, as well as their regularized variants, are popular robust methods to estimate the scatter or covariance matrix of a multivariate distribution. In this work, we study the non-asymptotic behavior of these estimators, for data sampled from a distribution that satisfies one of the following properties: 1) independent sub-Gaussian entries, up to a linear transformation; 2) log-concave distributions; 3) distributions satisfying a convex concentration property. Our main contribution is the derivation of tight non-asymptotic concentration bounds of these M-estimators around a suitably scaled version of the data sample covariance matrix. Prior to our work, nonasymptotic bounds were derived only for Elliptical and Gaussian distributions. Our proof uses a variety of tools from non asymptotic random matrix theory and high dimensional geometry. Finally, we illustrate the utility of our results on two examples of practical interest: sparse covariance and sparse precision matrix estimation.

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Section: Introductionmentioning

confidence: 92%

“…While the assumption that X has zero mean may not hold in practice, it has been used extensively in previous studies of Tyler's and Maronna's M-estimators, cf. [6,25,33,77]. In Section 6 we discuss how this restriction may be removed for some of our results.…”

mentioning

confidence: 99%

Tyler's and Maronna's M-estimators: Non-Asymptotic Concentration Results

Romanov¹,

Kur²,

Nadler³

2022

Preprint

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“…Relevant tools and promising directions can be leveraged from the fields of robust statistics, RMT, and high-dimensional covariance estimation (see, e.g. [15] and references therein).…”

Section: The Role Of Sp In the 6g Eramentioning

confidence: 99%

The Road to 6G: Ten Physical Layer Challenges for Communications Engineers

et al. 2021

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While the deployment of 5G cellular systems will continue well in to the next decade, much interest is already being generated towards technologies that will underlie its successor, 6G. Undeniably, 5G will have transformative impact on the way we live and communicate, yet, it is still far away from supporting the Internet-of-Everything (IoE), where upwards of a million devices per km 3 (both terrestrial and aerial) will require ubiquitous, reliable, low-latency connectivity. This article looks at some of the fundamental problems that pertain to key physical layer enablers for 6G. This includes highlighting challenges related to intelligent reflecting surfaces, cell-free massive MIMO and THz communications. Our analysis covers theoretical modeling challenges, hardware implementation issues and scalability among others. The article concludes by delineating the critical role of signal processing in the new era for wireless communications.

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“…This subsection presents an asymptotic estimator (AE), referred to as S 2 CM-AE, for the large (N, L) regime, which has a more concise expression. For identity targets, we will apply results from random matrix theory (RMT) [71]- [73] to demonstrate that S 2 CM-AE is equivalent to S 2 CM-CV in the asymptotic regime where both the dimensionality N and number of samples L approach infinity.…”

Section: B Asymptotic Estimatormentioning

confidence: 99%

“…Let us make the following statistical assumptions on the large-dimensional random matrices under study, unless otherwise stated [71]- [73]:…”

Section: B Asymptotic Estimatormentioning

confidence: 99%

Cross-Validated Tuning of Shrinkage Factors for MVDR Beamforming Based on Regularized Covariance Matrix Estimation

Xie¹,

He²,

Tong³

et al. 2021

Preprint

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This paper considers the regularized estimation of covariance matrices (CM) of high-dimensional (compound) Gaussian data for minimum variance distortionless response (MVDR) beamforming. Linear shrinkage is applied to improve the accuracy and condition number of the CM estimate for lowsample-support cases. We focus on data-driven techniques that automatically choose the linear shrinkage factors for shrinkage sample covariance matrix (S 2 CM) and shrinkage Tyler's estimator (STE) by exploiting cross validation (CV). We propose leaveone-out cross-validation (LOOCV) choices for the shrinkage factors to optimize the beamforming performance, referred to as S 2 CM-CV and STE-CV. The (weighted) out-of-sample output power of the beamfomer is chosen as a proxy of the beamformer performance and concise expressions of the LOOCV cost function are derived to allow fast optimization. For the large system regime, asymptotic approximations of the LOOCV cost functions are derived, yielding the S 2 CM-AE and STE-AE. In general, the proposed algorithms are able to achieve near-oracle performance in choosing the linear shrinkage factors for MVDR beamforming. Simulation results are provided for validating the proposed methods.

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Large-Dimensional Behavior of Regularized Maronna's M-Estimators of Covariance Matrices

Cited by 7 publications

References 35 publications

Tyler's and Maronna's M-estimators: Non-Asymptotic Concentration Results

Tyler's and Maronna's M-estimators: Non-Asymptotic Concentration Results

The Road to 6G: Ten Physical Layer Challenges for Communications Engineers

Cross-Validated Tuning of Shrinkage Factors for MVDR Beamforming Based on Regularized Covariance Matrix Estimation

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