Optimal transport (OT) theory describes general principles to define and select, among many possible choices, the most efficient way to map a probability measure onto another. That theory has been mostly used to estimate, given a pair of source and target probability measures $(\mu,\nu)$, a parameterized map $T_\theta$ that can efficiently map $\mu$ onto $\nu$. In many applications, such as predicting cell responses to treatments, the data measures $\mu,\nu$ (features of untreated/treated cells) that define optimal transport problems do not arise in isolation but are associated with a context $c$ (the treatment). To account for and incorporate that context in OT estimation, we introduce CondOT, an approach to estimate OT maps conditioned on a context variable, using several pairs of measures $(\mu_i, \nu_i)$ tagged with a context label $c_i$. Our goal is to % extract from a dataset of labeled pairs $\{(c_i, (\mu_i, \nu_i))\}$ learn a global map $\mathcal{T}_{\theta}$ which is not only expected to fit em all pairs in the dataset $\{(c_i, (\mu_i, \nu_i))\}$, i.e., $\mathcal{T}_{\theta}(c_i) \sharp\mu_i \approx \nu_i$, but should generalize to produce meaningful maps $\mathcal{T}_{\theta}(c_{\text{new}})$ conditioned on unseen contexts $c_{\text{new}}$. Our approach harnesses and provides a novel usage for partially input convex neural networks, for which we introduce a robust and efficient initialization strategy inspired by Gaussian approximations. We demonstrate the ability of CondOT to infer the effect of an arbitrary combination of genetic or therapeutic perturbations on single cells, using only observations of the effects of said perturbations separately.

使用CondOT，一种多任务学习方法，结合上下文变量和部分输入凸的神经网络，可以在给定单细胞的基因或治疗干预效果的情况下，推断出其任意组合的效果。

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