Given samples from two joint distributions, we consider the problem of Optimal Transportation (OT) between them when conditioned on a common variable. We focus on the general setting where the conditioned variable may be continuous, and the marginals of this variable in the two joint distributions may not be the same. In such settings, standard OT variants cannot be employed, and novel estimation techniques are necessary. Since the main challenge is that the conditional distributions are not explicitly available, the key idea in our OT formulation is to employ kernelized-least-squares terms computed over the joint samples, which implicitly match the transport plan's marginals with the empirical conditionals. Under mild conditions, we prove that our estimated transport plans, as a function of the conditioned variable, are asymptotically optimal. For finite samples, we show that the deviation in terms of our regularized objective is bounded by $O(1/m^{1/4})$, where $m$ is the number of samples. We also discuss how the conditional transport plan could be modelled using explicit probabilistic models as well as using implicit generative ones. We empirically verify the consistency of our estimator on synthetic datasets, where the optimal plan is analytically known. When employed in applications like prompt learning for few-shot classification and conditional-generation in the context of predicting cell responses to treatment, our methodology improves upon state-of-the-art methods.

翻译：给定来自两个联合分布的样本，我们考虑在共同变量条件下它们之间的最优传输问题。我们聚焦于一般场景，其中条件变量可能为连续型，且该变量在两个联合分布中的边缘分布可能不同。在此类设定下，标准最优传输变体无法适用，需要新的估计技术。由于主要挑战在于条件分布并非显式可得，我们最优传输公式的核心思想是利用联合样本上计算的核化最小二乘项，这些项隐式地将传输计划的边缘分布与经验条件分布相匹配。在温和条件下，我们证明所估计的传输计划作为条件变量的函数是渐近最优的。对于有限样本，我们证明正则化目标的偏差受限于$O(1/m^{1/4})$（其中$m$为样本数）。同时讨论如何通过显式概率模型和隐式生成模型对条件传输计划进行建模。我们在合成数据集上经验验证了估计量的一致性——这些数据集的已知解析最优传输计划。当应用于少样本分类的提示学习以及预测细胞对治疗反应的条件生成等场景时，我们的方法显著优于现有最优方法。