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.

翻译：我们研究非平衡最优输运（UOT）问题，其中边际约束通过最大均值差异（MMD）正则化来强制执行。本研究的动机源于观察到现有UOT工作主要基于$\phi$-散度（例如KL）进行正则化。属于积分概率度量（IPM）互补族的MMD作为UOT中正则化子的作用似乎尚未得到充分理解。我们的主要结果基于Fenchel对偶，通过该对偶能够研究MMD正则化UOT（MMD-UOT）的性质。该对偶结果的一个有趣结论是，MMD-UOT在测度上诱导出一种新型度量，该度量同样属于IPM族。此外，我们提出了基于有限样本的凸规划方法用于估计MMD-UOT及其对应的重心。在温和条件下，我们证明了基于凸规划的估计量具有一致性，且估计误差以$\mathcal{O}\left(m^{-\frac{1}{2}}\right)$速率衰减，其中$m$为源/目标测度的样本数。最后，我们讨论了如何通过（加速）投影梯度下降高效求解这些凸规划。通过多样化实验表明，MMD-UOT在机器学习应用中是有望替代$\phi$-散度正则化UOT的方法。