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 mild 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.

翻译：本文研究非平衡最优传输（UOT）问题，其中利用最大均值差异（MMD）正则化来施加边际约束。我们的研究动机源于观察到现有关于UOT的文献主要集中于基于$\phi$散度（如KL散度）的正则化。尽管MMD广受欢迎，但其在UOT框架中作为正则化项的作用尚待深入理解。我们首先推导出MMD正则化UOT（MMD-UOT）的一种特定对偶形式，这有助于证明若干有用性质。该对偶结果的一个有趣结论是：MMD-UOT不仅像Wasserstein距离那样提升基础度量结构，还像MMD那样在样本层面具有高效估计性。对于涉及非离散度量的实际应用，我们提出了一种仅依赖于给定（$m$）样本支撑的传输计划估计量。在温和条件下，我们证明该有限支撑传输计划的估计误差同样为$\mathcal{O}(1/\sqrt{m})$。据我们所知，这种免于维度灾难的误差界在$\phi$散度正则化UOT中尚未被报道。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，在多种机器学习应用中，MMD-UOT始终优于包括KL正则化UOT和MMD在内的主流基线方法。