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Spectral Shrinkage of Tyler's M -Estimator of Covariance Matrix

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Abstract

Covariance matrices usually exhibit specific spectral structures, such as low-rank ones in the case of factor models. In order to exploit this prior knowledge in a robust estimation process, we propose a new regularized version of Tyler's M-estimator of covariance matrix. This estimator is expressed as the minimizer of a robust M-estimating cost function plus a penalty that is unitary invariant (i.e., that only applies on the eigenvalue) that shrinks the estimated spectrum toward a fixed target. The structure of the estimate is expressed through an interpretable fixed-point equation. A majorization-minimization (MM) algorithm is derived to compute this estimator, and the g-convexity of the objective is also discussed. Several simulation studies illustrate the interest of the approach and also explore a method to automatically choose the target spectrum through an auxiliary estimator.
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Dates and versions

hal-02485823 , version 1 (20-02-2020)

Identifiers

  • HAL Id : hal-02485823 , version 1

Cite

Arnaud Breloy, Esa Ollila, Frédéric Pascal. Spectral Shrinkage of Tyler's M -Estimator of Covariance Matrix. IEEE CAMSAP 2019 - IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Dec 2019, Guadeloupe, West Indies, France. ⟨hal-02485823⟩
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