Robust regularized M-estimators of regression parameters and covariance matrix

High dimension low sample size (HD-LSS) data are becoming increasingly present in a variety of fields, including chemometrics and medical imaging. Especially problems with n < p (more variables than measurements) present a challenge to data analysts since the classical techniques can not be used. In this paper, we consider HD-LSS data in regression parameter and covariance matrix estimation problems. In particular, we consider and extend convex relaxation (or shrinkage regularization, diagonal loading) approach for M–estimation of regression coefficients and covariance (scatter) matrix. We demonstrate the utility of the methods in beamforming and tensor decomposition applications