  Optimal transport (OT) measures distances between distributions in a way that
depends on the geometry of the sample space. In light of recent advances in
computational OT, OT distances are widely used as loss functions in machine
learning. Despite their prevalence and advantages, OT loss functions can be
extremely sensitive to outliers. In fact, a single adversarially-picked outlier
can increase the standard $W_2$-distance arbitrarily. To address this issue, we
propose an outlier-robust formulation of OT. Our formulation is convex but
challenging to scale at a first glance. Our main contribution is deriving an
\emph{equivalent} formulation based on cost truncation that is easy to
incorporate into modern algorithms for computational OT. We demonstrate the
benefits of our formulation in mean estimation problems under the Huber
contamination model in simulations and outlier detection tasks on real data.
