  Optimal transport (OT) naturally arises in a wide range of machine learning
applications but may often become the computational bottleneck. Recently, one
line of works propose to solve OT approximately by searching the
\emph{transport plan} in a low-rank subspace. However, the optimal transport
plan is often not low-rank, which tends to yield large approximation errors.
For example, when Monge's \emph{transport map} exists, the transport plan is
full rank. This paper concerns the computation of the OT distance with adequate
accuracy and efficiency. A novel approximation for OT is proposed, in which the
transport plan can be decomposed into the sum of a low-rank matrix and a sparse
one. We theoretically analyze the approximation error. An augmented Lagrangian
method is then designed to efficiently calculate the transport plan.
