  Regularizing the optimal transport (OT) problem has proven crucial for OT
theory to impact the field of machine learning. For instance, it is known that
regularizing OT problems with entropy leads to faster computations and better
differentiation using the Sinkhorn algorithm, as well as better sample
complexity bounds than classic OT. In this work we depart from this practical
perspective and propose a new interpretation of regularization as a robust
mechanism, and show using Fenchel duality that any convex regularization of OT
can be interpreted as ground cost adversarial. This incidentally gives access
to a robust dissimilarity measure on the ground space, which can in turn be
used in other applications. We propose algorithms to compute this robust cost,
and illustrate the interest of this approach empirically.
