Supplementary Material: Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation

In this section, we present the proofs: Duality: Theorem 1. For brevity, we shall ignore explicitly stating it in the rest of the proof. Now, we write the Lagrangian function with respect to marginal constraints. Furthermore, when the distributions lie in a metric space, we can further simplify this duality. Since the feasible set in the dual problem satisfies φ(x) ψ(y) c(x, y), φ(x) k(x), and by using y = x in Eq (3), we obtain, k(x) ψ(x).