MMD-regularized Unbalanced Optimal Transport

We study the unbalanced optimal transport (UOT) problem, where the marginal constraints are enforced using Maximum Mean Discrepancy (MMD) regularization. Our study is motivated by the observation that existing works on UOT have mainly focused on regularization based on $\phi$-divergence (e.g., KL). The role of MMD, which belongs to the complementary family of integral probability metrics (IPMs), as a regularizer in the context of UOT seems to be less understood. Our main result is based on Fenchel duality, using which we are able to study the properties of MMD-regularized UOT (MMD-UOT). One interesting outcome of this duality result is that MMD-UOT induces a novel metric over measures, which again belongs to the IPM family. Further, we present finite-sample-based convex programs for estimating MMD-UOT and the corresponding barycenter. Under mild conditions, we prove that our convex-program-based estimators are consistent, and the estimation error decays at a rate $\mathcal{O}\left(m^{-\frac{1}{2}}\right)$, where $m$ is the number of samples from the source/target measures. Finally, we discuss how these convex programs can be solved efficiently using (accelerated) projected gradient descent. We conduct diverse experiments to show that MMD-UOT is a promising alternative to $\phi$-divergence-regularized UOT in machine learning applications.