  Optimal transport (OT) provides a way of measuring distances between
distributions that depends on the geometry of the sample space. In light of
recent advances in solving the OT problem, OT distances are widely used as loss
functions in minimum distance estimation. Despite its prevalence and
advantages, however, OT is extremely sensitive to outliers. A single
adversarially-picked outlier can increase OT distance arbitrarily. To address
this issue, in this work we propose an outlier-robust OT formulation. Our
formulation is convex but challenging to scale at a first glance. We proceed by
deriving an \emph{equivalent} formulation based on cost truncation that is easy
to incorporate into modern stochastic algorithms for regularized OT. We
demonstrate our model applied to mean estimation under the Huber contamination
model in simulation as well as outlier detection on real data.
