We study the unbalanced optimal transport (UOT) problem, where the marginal constraints are enforced using Maximum Mean Discrepancy (MMD) regularization. Our work is motivated by the observation that the literature on UOT is focused on regularization based on $\phi$-divergence (e.g., KL divergence). Despite the popularity of MMD, its role as a regularizer in the context of UOT seems less understood. We begin by deriving a specific dual of MMD-regularized UOT (MMD-UOT), which helps us prove several useful properties. One interesting outcome of this duality result is that MMD-UOT induces novel metrics, which not only lift the ground metric like the Wasserstein but are also sample-wise efficient to estimate like the MMD. Further, for real-world applications involving non-discrete measures, we present an estimator for the transport plan that is supported only on the given ($m$) samples. Under certain conditions, we prove that the estimation error with this finitely-supported transport plan is also $\mathcal{O}(1/\sqrt{m})$. As far as we know, such error bounds that are free from the curse of dimensionality are not known for $\phi$-divergence regularized UOT. Finally, we discuss how the proposed estimator can be computed efficiently using accelerated gradient descent. Our experiments show that MMD-UOT consistently outperforms popular baselines, including KL-regularized UOT and MMD, in diverse machine learning applications. Our codes are publicly available at https://github.com/Piyushi-0/MMD-reg-OT

翻译：我们研究了非平衡最优传输（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。我们的工作源于观察到关于UOT的文献主要集中在基于φ-散度（如KL散度）的正则化上。尽管MMD广受欢迎，但其在UOT中作为正则化器的作用似乎尚未得到充分理解。我们首先推导出MMD正则化UOT（MMD-UOT）的一个特定对偶形式，这有助于我们证明若干有用性质。该对偶结果的一个有趣结论是，MMD-UOT引入了新度量，这些度量不仅像Wasserstein距离一样提升底度量，而且像MMD一样在样本层面易于高效估计。此外，针对涉及非离散测度的实际应用，我们提出了一种仅支撑于给定（$m$）个样本上的传输计划估计量。在特定条件下，我们证明使用该有限支撑传输计划的估计误差也为$\mathcal{O}(1/\sqrt{m})$。据我们所知，这种无维度灾难的误差界在φ-散度正则化UOT中尚未见报道。最后，我们讨论了如何利用加速梯度下降法高效计算所提出的估计量。实验表明，MMD-UOT在多种机器学习应用中始终优于包括KL正则化UOT和MMD在内的主流基线方法。我们的代码已公开在 https://github.com/Piyushi-0/MMD-reg-OT。