Performance Analysis of Tyler's Covariance Estimator

Soloveychik, Ilya; Wiesel, Ami

doi:10.1109/tsp.2014.2376911

Cited by 32 publications

(16 citation statements)

References 48 publications

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“…Tyler and named after him Tyler (1987). Tyler's covariance estimator, given by formula (4), can be equivalently defined as a covariance parameter MLE of a certain spherical distribution Greco and Gini (2013); Soloveychik and Wiesel (2015) as follows.…”

Section: Tyler's Estimatormentioning

confidence: 99%

“…After normalization any GE vector becomes RACE distributed. This allows us to treat all these distributions together using Tyler's estimator, which is the MLE of the shape matrix parameter in RACE populations and is unbiased when a specific scaling is fixed Greco and Gini (2013); Soloveychik and Wiesel (2015).…”

Section: Tyler's Estimatormentioning

confidence: 99%

“…Therefore, Tyler's estimator has been successfully used to replace the SCM in many applications sensitive to outliers or heavy-tailed noise, such as anomaly detection in wireless sensor networks Chen et al (2011), antenna array processing Ollila and Koivunen (2003), and radar detection Abramovich et al (2007); Bandiera et al (2010); Ollila et al (2012); Pascal et al (2008). It was recently demonstrated that Tyler's estimator can be viewed as an MLE of a certain spherical distribution Greco and Gini (2013); Soloveychik and Wiesel (2015); Wiesel (2012a). In spite of the fact that the obtained MLE program is not convex, it was later shown to become convex under the geodesic metric change we mentioned above Wiesel (2012a,b).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness

Soloveychik

Trushin

2016

Journal of Multivariate Analysis

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We study the Gaussian and robust covariance estimation, assuming the true covariance matrix to be a Kronecker product of two lower dimensional square matrices. In both settings we define the estimators as solutions to the constrained maximum likelihood programs. In the robust case, we consider Tyler's estimator defined as the maximum likelihood estimator of a certain distribution on a sphere. We develop tight sufficient conditions for the existence and uniqueness of the estimates and show that in the Gaussian scenario with the unknown mean, p/q + q/p + 2 samples are almost surely enough to guarantee the existence and uniqueness, where p and q are the dimensions of the Kronecker product factors. In the robust case with the known mean, the corresponding sufficient number of samples is max [p/q, q/p] + 1.

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Section: Tyler's Estimatormentioning

confidence: 99%

Section: Tyler's Estimatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness

Soloveychik

Trushin

2016

Journal of Multivariate Analysis

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“…In this section we provide a high probability performance bound for the proposed JICE algorithm. In the derivation presented here we follow the technique used by [1], [29]- [31]. The method consists in showing that the extremal point of the target (22) lies within some small vicinity of the true parameter value with high probability.…”

Section: E Upper Performance Boundmentioning

confidence: 99%

Joint inverse covariances estimation with mutual linear structure

Soloveychik

Wiesel

2015

2015 23rd European Signal Processing Conference (EUSIPCO)

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We consider the problem of joint estimation of structured inverse covariance matrices. We perform the estimation using groups of measurements with different covariances of the same unknown structure. Assuming the inverse covariances to span a low dimensional linear subspace in the space of symmetric matrices, our aim is to determine this structure. It is then utilized to improve the estimation of the inverse covariances. We propose a novel optimization algorithm discovering and exploiting the underlying structure and provide its efficient implementation. Numerical simulations are presented to illustrate the performance benefits of the proposed algorithm.Index Terms-Structured inverse covariance estimation, joint inverse covariance estimation, graphical models. of these works is that they consider a single and static environment where the structure of the true concentration matrix, or at least the class of structures, as in the sparse case, is known in advance. Often, this is not the case and techniques are needed to learn the structure from the observations. A typical approach is to consider multiple datasets sharing a similar structure but non homogeneous environments [8]-[12]. This is, for example, the case in covariance estimation for classification across multiple classes [13]. A related problem addresses tracking a time varying covariance throughout a stream of data [14], [15], where it is assumed that the structure changes at a slower rate than the covariances themselves [16].Here too, it is natural to divide this stream of data into independent blocks of measurements.

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“…The elliptical distribution family contains a much broader class of distributions than the normal distribution family, such as mixture of normal distributions, multivariate t-distribution, multi-uniform distribution on unit sphere, Pearson Type II distribution, among others. It has been used as a tool to study the robustness of normality in the literature of multivariate nonparametric tests (Mottonen, Oja and Tienari, 1997; Oja and Randles, 2004; Chen, Wiesel and Hero, 2011; Soloveychik and Wiesel, 2015; Wang, Peng and Li, 2015). The elliptical linear regressions have been proposed in Osiewalski (1991); Osiewalski and Steel (1993); Arellano-Valle, del Pino and Iglesias (2006); Fan and Lv (2008); Liang and Li (2009); Vidal and Arellano-Valle (2010), and have received more and more attentions in the recent literature (Arellano-Valle, del Pino and Iglesias, 2006; Fan and Lv, 2008; Liang and Li, 2009; Vidal and Arellano-Valle, 2010).…”

Section: Introductionmentioning

confidence: 99%

Variable Selection via Partial Correlation

Liu

Lou

2018

STAT SINICA

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Partial correlation based variable selection method was proposed for normal linear regression models by Bühlmann, Kalisch and Maathuis (2010) as a comparable alternative method to regularization methods for variable selection. This paper addresses two important issues related to partial correlation based variable selection method: (a) whether this method is sensitive to normality assumption, and (b) whether this method is valid when the dimension of predictor increases in an exponential rate of the sample size. To address issue (a), we systematically study this method for elliptical linear regression models. Our finding indicates that the original proposal may lead to inferior performance when the marginal kurtosis of predictor is not close to that of normal distribution. Our simulation results further confirm this finding. To ensure the superior performance of partial correlation based variable selection procedure, we propose a thresholded partial correlation (TPC) approach to select significant variables in linear regression models. We establish the selection consistency of the TPC in the presence of ultrahigh dimensional predictors. Since the TPC procedure includes the original proposal as a special case, our theoretical results address the issue (b) directly. As a by-product, the sure screening property of the first step of TPC was obtained. The numerical examples also illustrate that the TPC is competitively comparable to the commonly-used regularization methods for variable selection.

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Performance Analysis of Tyler's Covariance Estimator

Cited by 32 publications

References 48 publications

Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness

Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness

Joint inverse covariances estimation with mutual linear structure

Variable Selection via Partial Correlation

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