  In several applications, including imaging of deformable objects while in
motion, simultaneous localization and mapping, and unlabeled sensing, we
encounter the problem of recovering a signal that is measured subject to
unknown permutations. In this paper we take a fresh look at this problem
through the lens of optimal transport (OT). In particular, we recognize that in
most practical applications the unknown permutations are not arbitrary but some
are more likely to occur than others. We exploit this by introducing a
regularization function that promotes the more likely permutations in the
solution. We show that, even though the general problem is not convex, an
appropriate relaxation of the resulting regularized problem allows us to
exploit the well-developed machinery of OT and develop a tractable algorithm.
