Given samples from two joint distributions, we consider the problem of
Optimal Transportation (OT) between the corresponding distributions conditioned
on a common variable. The objective of this work is to estimate the associated
transport cost (Wasserstein distance) as well as the transport plan between the
conditionals as a function of the conditioned value. Since matching conditional
distributions is at the core of supervised training of discriminative models
and (implicit) conditional-generative models, OT between conditionals has the
potential to be employed in diverse machine learning applications. However,
since the conditionals involved in OT are implicitly specified via the joint
samples, it is challenging to formulate this problem, especially when (i) the
variable conditioned on is continuous and (ii) the marginal of this variable in
the two distributions is different. We overcome these challenges by employing a
specific kernel MMD (Maximum Mean Discrepancy) based regularizer that ensures
the marginals of our conditional transport plan are close to the conditionals
specified via the given joint samples. Under mild conditions, we prove that our
estimator for this regularized transport cost is statistically consistent and
derive finite-sample bounds on the estimation error. Application-specific
details for parameterizing our conditional transport plan are also presented.
Furthermore, we empirically evaluate our methodology on benchmark datasets in
applications like classification, prompt learning for few-shot classification,
and conditional-generation in the context of predicting cell responses to
cancer treatment.

本文研究了基于公共变量的两个联合分布的条件 Optimal Transportation 问题，提出了一种基于 Maximum Mean Discrepancy 的正则化器用于解决连续变量和分布不同的情况，进而实现了条件转移计划的估计和统计一致性证明，并在分类、few-shot 分类和细胞响应预测等领域中进行了实证评估。