Parallel proximal decomposition algorithmsfor robust estimation

In the past few years, robustness has been one problem that was given much attention inthe statistical literature. While it is now clear that no single robust regression procedure isbest (by mean square error or other adequate criteria), the LAV (least absolute value) andthe Huber‐M estimators are currently attracting considerable attention when the errors havea contaminated Gaussian or long‐tail distribution. Finding efficient algorithms to producesuch estimates in the case of large data sets is still a field of active research. In this paper,we present algorithms based on the Spingarn Partial Inverse proximal approach that takesinto account both primal and dual aspects of the related optimization problems. They can beviewed as decomposition methods. Known to be always globally convergent, such an alternativeiterative approach leads to simple computational steps and updating rules. The resultis a highly parallel algorithm particularly attractive for large‐scale problems. Its efficientimplementation on a parallel computer architecture is described. Remedies are introducedto ensure efficiency in the case of models with less than full ranks. Numerical simulationsare considered and computational performance reported. Finally, we show how the methodallows for easy handling of general convex constraints on the primal variables. We discussin detail a variety of linear and nonlinear restrictions. The case of ridge LAV and Huber‐Mregression is specifically considered. Copyright Kluwer Academic Publishers 1999