Supervised Training of Conditional Monge Maps

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 (µ, υ), a parameterized map T_θ that can efficiently map µ onto υ. In many applications, such as predicting cell responses to treatments, pairs of input/output data measures (µ, υ) 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 (µ_i, υ_i) tagged with a context label c_i. CONDOT learns a global map T_θ conditioned on context that is not only expected to fit all labeled pairs in the dataset ((c_i, (µ_i, υ_i))), i.e., T_θ(c_i)]µ_i ≃ υ_i, but should also generalize to produce meaningful maps T_θ(c_new) when conditioned on unseen contexts c_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.