Joint outlier detection and variable selection using discrete optimization

In regression, the quality of estimators is known to be very sensitive to the presence of spurious variables and outliers. Unfortunately, this is a frequent situation when dealing with real data. To handle outlier proneness and achieve variable selection, we propose a robust method performing the outright rejection of discordant observations together with the selection of relevant variables. A natural way to define the corresponding optimization problem is to use the l(0) norm and recast it as a mixed integer optimization problem. To retrieve this global solution more efficiently, we suggest the use of additional constraints as well as a clever initialization. To this end, an efficient and scalable non-convex proximal alternate algorithm is introduced. An empirical comparison between the l(0) norm approach and its l(1) relaxation is presented as well. Results on both synthetic and real data sets provided that the mixed integer programming approach and its discrete first order warm start provide high quality solutions.