Unbalanced Optimal Transport using Integral Probability Metric Regularization

Unbalanced Optimal Transport (UOT) is the generalization of classical optimal transport to un-normalized measures. Existing formulations employ KL (or, in general, $\phi$-divergence) based regularization for handling the mismatch in the total mass of the given measures. Motivated by the known advantages of Integral Probability Metrics (IPMs) over $\phi$-divergences, this paper proposes novel formulations for UOT that employ IPMs for regularization. Under mild conditions, we show that the proposed formulations lift the ground metric to metrics over measures. More interestingly, the induced metric turns out to be another IPM whose generating set is the intersection of that of the IPM employed and the set of 1-Lipschitz functions under the ground metric. We further generalize these metrics to obtain analogues of p-Wasserstein metrics for the unbalanced case. When the regularizing IPM is chosen as the MMD, the proposed UOT as well as the corresponding Barycenter formulations, turn out to be those of minimizing a convex quadratic subject to non-negativity constraints and hence can be solved very efficiently. We also discuss connections with formulations for robust optimal transport in the case of noisy marginals.