On optimal transport with f-divergence regularization

This report establishes that an optimal transport (OT) problem regularized by a given f-divergence admits the same solution as another OT problem regularized by a different gdivergence, under an appropriate transformation of the cost function. This structural equivalence between OT problems regularized by distinct divergences, in the sense of sharing the same unique minimizer, is demonstrated within the framework of polish spaces with bounded cost functions. Under these general assumptions, the existence of optimal potentials for the dual problem and the uniqueness of the optimal coupling for the primal problem are rigorously established.