  Optimal Transport, a theory for optimal allocation of resources, is widely
used in various fields such as astrophysics, machine learning, and imaging
science. However, many applications impose elementwise constraints on the
transport plan which traditional optimal transport cannot enforce. Here we
introduce Supervised Optimal Transport (sOT) that formulates a constrained
optimal transport problem where couplings between certain elements are
prohibited according to specific applications. sOT is proved to be equivalent
to an $l^1$ penalized optimization problem, from which efficient algorithms are
designed to solve its entropy regularized formulation. We demonstrate the
capability of sOT by comparing it to other variants and extensions of
traditional OT in color transfer problem. We also study the barycenter problem
in sOT formulation, where we discover and prove a unique reverse and portion
selection (control) mechanism. Supervised optimal transport is broadly
applicable to applications in which constrained transport plan is involved and
the original unit should be preserved by avoiding normalization.
