Conditional Optimal Transport (COT) problem aims to find a transport map between conditional source and target distributions while minimizing the transport cost. Recently, these transport maps have been utilized in conditional generative modeling tasks to establish efficient mappings between the distributions. However, classical COT inherits a fundamental limitation of optimal transport, i.e., sensitivity to outliers, which arises from the hard distribution matching constraints. This limitation becomes more pronounced in a conditional setting, where each conditional distribution is estimated from a limited subset of data. To address this, we introduce the Conditional Unbalanced Optimal Transport (CUOT) framework, which relaxes conditional distribution-matching constraints through Csiszár divergence penalties while strictly preserving the conditioning marginals. We establish a rigorous formulation of the CUOT problem and derive its dual and semi-dual formulations. Based on the semi-dual form, we propose Conditional Unbalanced Optimal Transport Maps (CUOTM), an outlier-robust conditional generative model built upon a triangular $c$-transform parameterization. We theoretically justify the validity of this parameterization by proving that the optimal triangular map satisfies the $c$-transform relationships. Our experiments on 2D synthetic and image-scale datasets demonstrate that CUOTM achieves superior outlier robustness and competitive distribution-matching performance compared to existing COT-based baselines, while maintaining high sampling efficiency.

翻译：条件最优传输（COT）问题旨在寻找条件源分布与目标分布之间的传输映射，同时最小化传输代价。近年来，此类传输映射已被应用于条件生成建模任务，以建立分布间的高效映射。然而，经典COT继承了最优传输的一个根本局限，即对离群点的敏感性，这源于严格的分布匹配约束。该局限在条件设定下更为突出，因为每个条件分布仅从有限的数据子集进行估计。为解决此问题，我们引入了条件不平衡最优传输（CUOT）框架，该框架通过Csiszár散度惩罚松弛条件分布匹配约束，同时严格保持条件边缘分布。我们建立了CUOT问题的严格表述，并推导出其对偶与半对偶形式。基于半对偶形式，我们提出了条件不平衡最优传输映射（CUOTM），这是一种基于三角$c$-变换参数化的离群点鲁棒条件生成模型。我们通过证明最优三角映射满足$c$-变换关系，从理论上论证了该参数化的有效性。在二维合成数据集与图像级数据集上的实验表明，相较于现有的基于COT的基线方法，CUOTM在保持高采样效率的同时，实现了更优的离群点鲁棒性与具有竞争力的分布匹配性能。