Regularized $M$ -Estimators of Scatter Matrix

In this paper, a general class of regularized M-estimators of scatter matrix are proposed that are suitable also for low or insufficient sample support (small n and large p) problems. The considered class constitutes a natural generalization of M-estimators of scatter matrix (Maronna, 1976) and are defined as a solution to a penalized M-estimation cost function. Using the concept of geodesic convexity, we prove the existence and uniqueness of the regularized M-estimators of scatter and the existence and uniqueness of the solution to the corresponding M-estimating equations under general conditions. Unlike the non-regularized M-estimators of scatter, the regularized estimators are shown to exist for any data configuration. An iterative algorithm with proven convergence to the solution of the regularized M-estimating equation is also given. Since the conditions for uniqueness do not include the regularized versions of Tyler´s M-estimator, necessary and sufficient conditions for their uniqueness are established separately. For the regularized Tyler´s M-estimators, we also derive a simple, closed form, and data-dependent solution for choosing the regularization parameter based on shape matrix matching in the mean-squared sense. Finally, some simulations studies illustrate the improved accuracy of the proposed regularized M-estimators of scatter compared to their non-regularized counterparts in low sample support problems. An example of radar detection using normalized matched filter (NMF) illustrate that an adaptive NMF detector based on regularized M-estimators are able to maintain accurately the preset CFAR level.