Huber's M-Estimator on Underdetermined Problems

Abstract

After surveying the theoretical aspects of Huber's $M$-estimator on underdetermined problems, two finite algorithms are presented. Both proceed in a constructive manner by moving from one partition to an adjacent one. One of the algorithm, which uses the tuning constant as a continuation parameter, also has the facility to simultaneously estimate the tuning constant and scaling factor. Stable and efficient implementation of the algorithms is presented together with numerical results. The $L_1$-norm problem is mentioned as a special case.