Intrinsic Cramér–Rao Bounds for Scatter and Shape Matrices Estimation in CES Distributions

Breloy, Arnaud; Ginolhac, Guillaume; Renaux, Alexandre; Bouchard, Florent

doi:10.1109/lsp.2018.2886700

Cited by 25 publications

(28 citation statements)

References 40 publications

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“…of y n . The criterion J Rm, R (µ) is strongly related to the Fisher information metric derived for CES distributions in [32]. Using the relations linking the trace and the vec-operator [33, Tables 2 and 3], we rewrite (13) as…”

Section: Algorithm 1 Sesamementioning

confidence: 99%

Robust estimation of structured scatter matrices in (mis)matched models

et al. 2019

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Covariance matrix estimation is a ubiquitous problem in signal processing. In most modern signal processing applications, data are generally modeled by non-Gaussian distributions with covariance matrices exhibiting a particular structure. Taking into account this structure and the non-Gaussian behavior improve drastically the estimation accuracy. In this paper, we consider the estimation of structured scatter matrix for complex elliptically distributed observations, where the assumed model can differ from the actual distribution of the observations. Specifically, we tackle this problem, in a mismatched framework, by proposing a novel estimator, named StructurEd ScAtter Matrix Estimator (SESAME), which is based on a two-step estimation procedure. We conduct theoretical analysis on the unbiasedness and the asymptotic efficiency and Gaussianity of SESAME. In addition, we derive a recursive estimation procedure that iteratively applies the SESAME method, called Recursive-SESAME (R-SESAME), reaching with improved performance at lower sample support the (Mismatched) Cramér-Rao Bound. Furthermore, we show that some special cases of the proposed method allow to retrieve preexisting methods. Finally, numerical results corroborate the theoretical analysis and assess the usefulness of the proposed algorithms.

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Section: Algorithm 1 Sesamementioning

confidence: 99%

Robust estimation of structured scatter matrices in (mis)matched models

et al. 2019

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“…The part of the metric that concerns U is the so-called canonical metric on Stiefel [15], which is obtained by treating St p,k as the quotient U p /U p−k . The one that concerns Σ corresponds to a class of affine invariant metrics on H ++ k that are of interest when dealing with elliptical distributions as they are related to the Fisher information metric [16] 1 . It is readily checked that the metric (3) is invariant along the equivalence classes (2), i.e., for all…”

Section: Riemannian Geometrymentioning

confidence: 99%

“…where λ j is the j th eigenvalue of R −1R . With α = p+d p+d+1 and β = α−1, it is the distance on H ++ p corresponding to the Fisher metric of the multivariate Student t-distribution [16]. The second one, which measures the error between the true subspace span(U ) of R = I p + U ΣU H and its estimate 2 These α and β correspond to the parameters of the Fisher information metric for the multivariate Student t-distribution in H ++ p (cf.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The second one, which measures the error between the true subspace span(U ) of R = I p + U ΣU H and its estimate 2 These α and β correspond to the parameters of the Fisher information metric for the multivariate Student t-distribution in H ++ p (cf. [16] for details). However, notice that these parameters do not influence the obtained results in this context.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Riemannian Framework for Robust Covariance Matrix Estimation in Spiked Models

Bouchard¹,

Breloy²,

Ginolhac³

et al. 2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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This paper aims at providing an original Riemannian geometry to derive robust covariance matrix estimators in spiked models (i.e. when the covariance matrix has a low-rank plus identity structure). The considered geometry is the one induced by the product of the Stiefel manifold and the manifold of Hermitian positive definite matrices, quotiented by the unitary group. One of the main contributions is to consider a Riemannian metric related to the Fisher information metric of elliptical distributions, leading to new representations for the tangent spaces and a new retraction. A new robust covariance matrix estimator is then obtained as the minimizer of Tyler's cost function, redefined directly on the set of low-rank plus identity matrices, and computed with the aforementioned tools. The main interest of this approach is that it appears well suited to the cases where the sample size is lower than the dimension, as illustrated by numerical experiments.

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“…To overcome this issue, we propose to derive a new inequality by using an original error measure and the corresponding so-called intrinsic Cramér-Rao bound theoretically defined in [15] which fits well with the source separation problem. Inspired by the works of [16] and using the procedure in [17], the estimation error is measured by a new Riemannian distance. The associated Fisher information matrix is obtained from the Fisher information metric and orthonormal bases of the tangent spaces of the parameter manifold.…”

Section: Introductionmentioning

confidence: 99%

Riemannian Geometry and Cramér-rao Bound for Blind Separation of Gaussian Sources

Bouchard

Breloy

Renaux

et al. 2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We consider the optimal performance of blind separation of Gaussian sources. In practice, this estimation problem is solved by a two-step procedure: estimation of a set of covariance matrices from the observed data and approximate joint diagonalization of this set to find the unmixing matrix. Rather than studying the theoretical performance of a specific method, we are interested in the optimal attainable performance of any estimator. To do so, we consider the so-called intrinsic Cramér-Rao bound, which exploits the geometry of the parameters of the model. Unlike previous works developing a Cramér-Rao bound in this context, our solution does not require any additional hypotheses. To obtain our bound, we define and study a new Riemannian manifold holding the parameters of interest. An original estimation error measure is defined with the help of our Riemannian distance function. The corresponding Fisher information matrix is then obtained from the Fisher information metric and orthonormal bases on the tangent spaces of the manifold. Finally, our theoretical results are validated on simulated data.

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Intrinsic Cramér–Rao Bounds for Scatter and Shape Matrices Estimation in CES Distributions

Cited by 25 publications

References 40 publications

Robust estimation of structured scatter matrices in (mis)matched models

Robust estimation of structured scatter matrices in (mis)matched models

Riemannian Framework for Robust Covariance Matrix Estimation in Spiked Models

Riemannian Geometry and Cramér-rao Bound for Blind Separation of Gaussian Sources

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