Empirical Optimal Transport between Conditional Distributions

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.