Generalized Robust Shrinkage Estimator and Its Application to STAP Detection Problem

Abstract

Abstract-Recently, in the context of covariance matrix estimation, in order to improve as well as to regularize the performance of the Tyler's estimator [1] also called the FixedPoint Estimator (FPE) [2], a "shrinkage" fixed-point estimator has been originally introduced in [3]. First, this work extends the results of [4], [5] by giving the general solution of the "shrinkage" fixed-point algorithm. Secondly, by analyzing this solution, called the generalized robust shrinkage estimator, we prove that this solution converges to a unique solution when the shrinkage parameter β (losing factor) tends to 0. This solution is exactly the FPE with the trace of its inverse equal to the dimension of the problem. This general result allows one to give another interpretation of the FPE and more generally, on the Maximum Likelihood approach for covariance matrix estimation when constraints are added. Then, some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor. Finally, this work is applied to a Space-Time Adaptive Processing (STAP) detection problem on real STAP data.

Regularized <formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula>-Estimators of Scatter Matrix

“…the regularized Tyler's M -estimator with β = 1 − α. This existence and uniqueness of the regularized Tyler's M -estimator for this case, i.e., when β = 1 − α, has also been established in [21], but only under the condition that the data are in general position and hence Conditions A and B are automatically satisfied for such samples. A related regularized M -Tyler's estimator is given in [3] as the limit of the algorithm…”

“…Condition (8) implies Tr(Σ −1 ) = p(1 − β)/α and hence the choice β = 1 is excluded. If we choose β = 1 − α above, then the estimator Σ satisfies the constraint Tr(Σ −1 ) = p. Hereafter, when using this estimator, we assume without loss of generality that n * = n. This case β = 1 − α has been previously studied in [21].…”

“…Our general conditions do not apply to the cost function corresponding to Tyler's [24] M -estimator and hence this estimator is treated seperately, with necessary and sufficient conditions being established to ensure the uniqueness of solution for the penalized Tyler's cost function. Regularized versions of Tyler's M -estimator have also been recently studied in [21] for the case β = 1 − α and under more strict conditions on the sample, and also in [3], but not in the context as a solution to a penalized M -estimation cost function. Estimation of the regularization parameters using the expected likelihood approach was proposed in [1], [2] for the regularized Tyler's M -estimator of [3], whereas [5] based their analysis on random matrix theory (both n and p are large).…”

Numerical performances of low rank stap based on different heterogeneous clutter subspace estimators

“…• The Shrinkage-FPE (SFPE), also known as DiagonnalyLoaded FPE [16][17] [18], defined for β ∈]0, 1] by the fixed point equation:…”

The outlier-corrected-data-adaptive Lasso: A new robust estimator for the independent contamination model

“…When dealing with real-world data, the presence of impulsive noise and outliers [3], [4], [5], [6], [7] must also be accounted for. Recently, the MM-Lasso and adaptive MMLasso were introduced to robustify against outliers [8].…”