  The best subset selection (or "best subsets") estimator is a classic tool for
sparse regression, and developments in mathematical optimization over the past
decade have made it more computationally tractable than ever. Notwithstanding
its desirable statistical properties, the best subsets estimator is susceptible
to outliers and can break down in the presence of a single contaminated data
point. To address this issue, a robust adaption of best subsets is proposed
that is highly resistant to contamination in both the response and the
predictors. The adapted estimator generalizes the notion of subset selection to
both predictors and observations, thereby achieving robustness in addition to
sparsity. This procedure, referred to as "robust subset selection" (or "robust
subsets"), is defined by a combinatorial optimization problem for which modern
discrete optimization methods are applied. The robustness of the estimator in
terms of the finite-sample breakdown point of its objective value is formally
established. In support of this result, experiments on synthetic and real data
are reported that demonstrate the superiority of robust subsets over best
subsets in the presence of contamination. Importantly, robust subsets fares
competitively across several metrics compared with popular robust adaptions of
continuous shrinkage estimators.
