Unbalanced optimal transport (UOT) provides a flexible way to match or compare nonnegative finite Radon measures. However, UOT requires a predefined ground transport cost, which may misrepresent the data's underlying geometry. Choosing such a cost is particularly challenging when datasets live in heterogeneous spaces, often motivating practitioners to adopt Gromov-Wasserstein formulations. To address this challenge, we introduce cost-regularized unbalanced optimal transport (CR-UOT), a framework that allows the ground cost to vary while allowing mass creation and removal. We show that CR-UOT incorporates unbalanced Gromov-Wasserstein type problems through families of inner-product costs parameterized by linear transformations, enabling the matching of measures or point clouds across Euclidean spaces. We develop algorithms for such CR-UOT problems using entropic regularization and demonstrate that this approach improves the alignment of heterogeneous single-cell omics profiles, especially when many cells lack direct matches.

翻译：非平衡最优传输（UOT）为匹配或比较非负有限拉东测度提供了一种灵活的方法。然而，UOT需要预先定义基础传输成本，这可能无法准确反映数据的底层几何结构。当数据集存在于异构空间时，选择合适的成本尤为困难，这常常促使实践者采用Gromov-Wasserstein 公式。为解决这一挑战，我们引入了成本正则化非平衡最优传输（CR-UOT）框架，该框架允许基础传输成本变化，同时支持质量的生成与移除。我们证明，通过由线性变换参数化的内积成本族，CR-UOT能够涵盖非平衡Gromov-Wasserstein 类型问题，从而实现跨欧几里得空间的测度或点云匹配。我们利用熵正则化为此类CR-UOT问题开发了算法，并证明该方法能够提升异构单细胞组学谱的对齐效果，尤其是在大量细胞缺乏直接匹配的情况下。