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, pairs of
input/output data measures $(\mu, \nu)$ that define optimal transport problems
do not arise in isolation but are associated with a context $c$, as for
instance a treatment when comparing populations of untreated and treated cells.
To account for that context in OT estimation, we introduce CondOT, a multi-task
approach to estimate a family of OT maps conditioned on a context variable,
using several pairs of measures $\left(\mu_i, \nu_i\right)$ tagged with a
context label $c_i$. CondOT learns a global map $\mathcal{T}_\theta$
conditioned on context that is not only expected to fit all labeled pairs in
the dataset $\left\{\left(c_i,\left(\mu_i, \nu_i\right)\right)\right\}$, i.e.,
$\mathcal{T}_\theta\left(c_i\right) \sharp \mu_i \approx \nu_i$, but should
also generalize to produce meaningful maps $\mathcal{T}_\theta\left(c_{\text
{new }}\right)$ when 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。