Parallel proximal decomposition algorithms for robust estimation

In the past few years, robustness has been one problem that was given much attention in the statistical literature. While it is now clear that no single robust regression procedure is best (by mean square error or other adequate criteria), the LAV (least absolute value) and the Huber-M estimators are currently attracting considerable attention when the errors have a contaminated Gaussian or long-tail distribution. Finding efficient algorithms to produce such 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 takes into account both primal and dual aspects of the related optimization problems. They can be viewed as decomposition methods. Known to be always globally convergent, such an alternative iterative approach leads to simple computational steps and updating rules. The result is a highly parallel algorithm particularly attractive for large-scale problems. Its efficient implementation on a parallel computer architecture is described. Remedies are introduced to ensure efficiency in the case of models with less than full ranks. Numerical simulations are considered and computational performance reported. Finally, we show how the method allows for easy handling of general convex constraints on the primal variables. We discuss in detail a variety of linear and nonlinear restrictions. The case of ridge LAV and Huber-M regression is specifically considered.