  Optimal Transport is a useful metric to compare probability distributions and
to compute a pairing given a ground cost. Its entropic regularization variant
(eOT) is crucial to have fast algorithms and reflect fuzzy/noisy matchings.
This work focuses on Inverse Optimal Transport (iOT), the problem of inferring
the ground cost from samples drawn from a coupling that solves an eOT problem.
It is a relevant problem that can be used to infer unobserved/missing links,
and to obtain meaningful information about the structure of the ground cost
yielding the pairing. On one side, iOT benefits from convexity, but on the
other side, being ill-posed, it requires regularization to handle the sampling
noise. This work presents an in-depth theoretical study of the l1
regularization to model for instance Euclidean costs with sparse interactions
between features. Specifically, we derive a sufficient condition for the robust
recovery of the sparsity of the ground cost that can be seen as a far reaching
generalization of the Lasso's celebrated Irrepresentability Condition. To
provide additional insight into this condition, we work out in detail the
Gaussian case. We show that as the entropic penalty varies, the iOT problem
interpolates between a graphical Lasso and a classical Lasso, thereby
establishing a connection between iOT and graph estimation, an important
problem in ML.
