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The Random Matrix Regime of Maronna's M-estimator with elliptically distributed samples [article]

Romain Couillet, Frédéric Pascal, Jack W. Silverstein
2013 arXiv   pre-print
This article demonstrates that the robust scatter matrix estimator Ĉ_N∈ C^N× N of a multivariate elliptical population x_1,...  ...  ,x_n∈ C^N originally proposed by Maronna in 1976, and defined as the solution (when existent) of an implicit equation, behaves similar to a well-known random matrix model in the limiting regime where the  ...  In this article, we revisit the study of Maronna's estimator for elliptically distributed samples using a probabilistic approach (as opposed to the statistical approach used classically in robust estimation  ... 
arXiv:1311.7034v1 fatcat:66hbgjl7tbfvrlphmafdhzr744

The random matrix regime of Maronna's M-estimator with elliptically distributed samples

Romain Couillet, Frédéric Pascal, Jack W. Silverstein
2015 Journal of Multivariate Analysis  
solution (when existent) of an implicit equation, behaves similar to a well-known random matrix model in the limiting regime where the population N and sample n sizes grow at the same speed.  ...  This article demonstrates that the robust scatter matrix estimatorĈ N ∈ C N ×N of a multivariate elliptical population x 1 , . . . , x n ∈ C N originally proposed by Maronna in 1976, and defined as the  ...  In this article, we revisit the study of Maronna's estimator for elliptically distributed samples using a probabilistic approach (as opposed to the statistical approach used classically in robust estimation  ... 
doi:10.1016/j.jmva.2015.02.020 fatcat:xiiueijzo5hcvi2nqm3a6f7nca

Large dimensional analysis of Maronna's M-estimator with outliers

David Morales-Jimenez, Romain Couillet, Matthew R. McKay
2015 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)  
well-known random matrix model in the limiting regime where both the population and sample sizes grow to infinity at the same speed.  ...  Building on recent results in the random matrix analysis of robust estimators of scatter, we show that a certain class of such estimators obtained from samples containing outliers behaves similar to a  ...  Then, as N, n → ∞, 1 We could have considered samples with elliptical-like distributions instead but, in order not to confuse messages, we only characterize here the behavior of Maronna's estimator for  ... 
doi:10.1109/icassp.2015.7178605 dblp:conf/icassp/Morales-Jimenez15 fatcat:xtjl6isqf5ditkvbmkwkxkygta

Marčenko–Pastur law for Tyler's M-estimator

Teng Zhang, Xiuyuan Cheng, Amit Singer
2016 Journal of Multivariate Analysis  
We also extend this result to elliptical-distributed data samples for Tyler's M-estimator and non-isotropic Gaussian data samples for Maronna's M-estimator.  ...  This paper studies the limiting behavior of Tyler's and Maronna's Mestimators, in the regime that the number of samples n and the dimension p both go to infinity, and p/n converges to a constant y with  ...  Singer was partially supported by Award Number FA9550-12-1-0317 and FA9550-13-1-0076 from AFOSR, by Award Number R01GM090200 from the NIGMS, and by Award Number LTR DTD 06-05-2012 from the Simons Foundation  ... 
doi:10.1016/j.jmva.2016.03.010 fatcat:aun3zjwc2ba2zmnomgbdzmsq2y

Marchenko-Pastur Law for Tyler's M-estimator [article]

Teng Zhang, Xiuyuan Cheng, Amit Singer
2016 arXiv   pre-print
This paper studies the limiting behavior of Tyler's M-estimator for the scatter matrix, in the regime that the number of samples n and their dimension p both go to infinity, and p/n converges to a constant  ...  As a result, the spectral distribution of Tyler's M-estimator converges weakly to the Marčenko-Pastur distribution.  ...  Singer was partially supported by Award Number FA9550-12-1-0317 and FA9550-13-1-0076 from AFOSR, by Award Number R01GM090200 from the NIGMS, and by Award Number LTR DTD 06-05-2012 from the Simons Foundation  ... 
arXiv:1401.3424v4 fatcat:za4xiyd4gjbubpk4gzptbniln4

New model order selection in large dimension regime for complex elliptically symmetric noise

Eugenie Terreaux, Jean-Philippe Ovarlez, Frederic Pascal
2017 2017 25th European Signal Processing Conference (EUSIPCO)  
(Random Matrix Theory) when the noise environment is modelled by Complex Elliptically Symmetric (CES) distribution, with unknown scatter matrix.  ...  The proposed method consists first in estimating the Toeplitz structure of the background covariance matrix.  ...  ACKNOWLEDGMENT The authors would like to thank DGA (Ministry of Defense) for its financial support.  ... 
doi:10.23919/eusipco.2017.8081376 dblp:conf/eusipco/TerreauxO017 fatcat:zecxenq2tnczrgsjtvtsg67ama

On the Convergence of Maronna's $M$-Estimators of Scatter

Yacine Chitour, Romain Couillet, Frederic Pascal
2015 IEEE Signal Processing Letters  
These results find important implications in recent works on the large dimensional (random matrix) regime of robust M-estimation.  ...  In this paper, we propose an alternative proof for the uniqueness of Maronna's M-estimator of scatter (Maronna, 1976) for N vector observations y_1,..., y_N∈ R^m under a mild constraint of linear independence  ...  With the recent renewed interest in robust M -estimation under the random matrix regime N, m → ∞ with c N → c ∞ ∈ (0, 1) [6]- [9] , alternative proofs of existence and uniqueness have appeared motivated  ... 
doi:10.1109/lsp.2014.2367547 fatcat:mnpdvfqswrhypnwx4rs3i6i3ye

Robust M-estimator of scatter for large elliptical samples

Romain Couillet, Frederic Pascal
2014 2014 IEEE Workshop on Statistical Signal Processing (SSP)  
It is shown that a certain family of robust scatter estimators of elliptical samples behaves similar to a well-known random matrix model in the limiting regime where both the population N and sample n  ...  This result allows us to understand the structure of such estimators and in particular to derive their limiting eigenvalue distributions.  ...  Fig. 2 . 2 Histogram of the eigenvalues of 1 n n i=1 xix * i for n = 2500, N = 500, CN = diag(I125, 3I125, 10I250), τ1 with Γ(.5, 2)-distribution. of the eigenvalues of the sample covariance matrix same  ... 
doi:10.1109/ssp.2014.6884560 dblp:conf/ssp/Couillet014 fatcat:duqiwov2xna4vodfaanbg42f6y

Robust covariance estimation and linear shrinkage in the large dimensional regime

Romain Couillet, Matthew McKay
2014 2014 IEEE International Workshop on Machine Learning for Signal Processing (MLSP)  
We consider here the case of i.i.d. elliptical zero mean samples in the regime where both sample and population sizes are large.  ...  The article studies two regularized robust estimators of scatter matrices proposed in parallel in [1] and [2], based on Tyler's robust M-estimator [3] and on Ledoit and Wolf's shrinkage covariance matrix  ...  CONCLUDING REMARKS The article shows that, in the large dimensional random matrix regime, the Abramovich-Pascal and Chen estimators for elliptical samples x 1 , . . . , x n are (up to a variable change  ... 
doi:10.1109/mlsp.2014.6958867 dblp:conf/mlsp/CouilletM14 fatcat:gynel7ohjncthnl7huo7pjouau

New Model Order Selection In Large Dimension Regime For Complex Elliptically Symmetric Noise

Jean Philippe Ovarlez, Frédéric Pascal, Eugenie Terreaux
2018 Zenodo  
Publication in the conference proceedings of EUSIPCO, Kos island, Greece, 2017  ...  ACKNOWLEDGMENT The authors would like to thank DGA (Ministry of Defense) for its financial support.  ...  N → ∞, m → ∞ with the constant regime c N = m N → c, c > 0.  ... 
doi:10.5281/zenodo.1159233 fatcat:ntwauryey5ex7iqsbif2yiudhq

On the asymptotics of Maronna's robust PCA [article]

Gordana Draskovic and Arnaud Breloy and Frederic Pascal
2018 arXiv   pre-print
the EVD of M-estimators.  ...  in the context of complex elliptically symmetric distributions.  ...  Their analysis in the large random matrix regime (i.e. when both the number of samples and the dimension tends to infinity at the same rate) has been established in [13] , [14] .  ... 
arXiv:1811.02069v1 fatcat:ccoaiwqmcngrhfk42jhrn65u64

A Toeplitz-Tyler Estimation of the Model Order in Large Dimensional Regime

Eugenie Terreaux, Jean-Philippe Ovarlez, Frederic Pascal
2018 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)  
This paper presents a new algorithm to estimate the number of sources embedded in a correlated Complex Elliptically Distributed (CES) noise in the context of large dimensional regime.  ...  The proposed method is a two-steps ones: first the data covariance matrix is estimated with a robust and consistent estimator exploiting the Toeplitz structure assumption of the true scatter matrix.  ...  Moreover, the m × p matrix M with elements M i,j = (M) i,j = (m j ) i is referred to as the mixing matrix and contains the p vectors of the sources.  ... 
doi:10.1109/icassp.2018.8461915 dblp:conf/icassp/TerreauxO018 fatcat:kxxw5zloofagjjv3toevjcrye4

Robust shrinkage M-estimators of large covariance matrices

Nicolas Auguin, David Morales-Jimenez, Matthew McKay, Romain Couillet
2016 2016 IEEE Statistical Signal Processing Workshop (SSP)  
By applying tools from random matrix theory, we characterize the asymptotic performance of such estimators when the number of samples and variables grow large together.  ...  Robust high dimensional covariance estimators are considered, comprising regularized (linear shrinkage) modifications of Maronna's classical M-estimators.  ...  heavy-tailed distributions, e.g. elliptical data [13] .  ... 
doi:10.1109/ssp.2016.7551720 dblp:conf/ssp/AuguinMMC16 fatcat:yqfrendbcfarjexddgl5iu4e3m

Large Dimensional Analysis of Robust M-Estimators of Covariance With Outliers

David Morales-Jimenez, Romain Couillet, Matthew R. McKay
2015 IEEE Transactions on Signal Processing  
Building upon recent random matrix advances in the area of robust statistics, we specifically show that the so-called Maronna M-estimator of scatter asymptotically behaves similar to well-known random  ...  Thus, robust M-estimators can bring substantial benefits over more simplistic estimators such as the per-sample normalized version of the sample covariance matrix, which is not capable of differentiating  ...  Most investigations of robust estimators of scatter focus on the more tractable case where the samples (i.e., the columns of Y) are independent with identical elliptical distribution.  ... 
doi:10.1109/tsp.2015.2460225 fatcat:surjulxusjhzrm2hateqzyjwlq

Robust Sparse Covariance Estimation by Thresholding Tyler's M-Estimator [article]

John Goes, Gilad Lerman, Boaz Nadler
2018 arXiv   pre-print
Towards bridging this gap, in this work we consider estimating a sparse shape matrix from n samples following a possibly heavy tailed elliptical distribution.  ...  We derive bounds on the difference in spectral norm between our estimators and the shape matrix in the joint limit as the dimension p and sample size n tend to infinity with p/n→γ>0.  ...  An important property of the elliptical distribution is that if x 1 , x 2 are independent random vectors from (1), then x 1 − x 2 has an elliptical distribution with the same shape matrix S p but with  ... 
arXiv:1706.08020v3 fatcat:fqerztgov5fmppztaekja37biq
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