Supervised Training of Conditional Monge Maps
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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,ν), a parameterized map T_θ that can efficiently map μ onto ν. In many applications, such as predicting cell responses to treatments, the data measures μ,ν (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 (μ_i, ν_i) tagged with a context label c_i. Our goal is to pairs {(c_i, (μ_i, ν_i))} learn a global map 𝒯_θ which is not only expected to fit em all pairs in the dataset {(c_i, (μ_i, ν_i))}, i.e., 𝒯_θ(c_i) ♯μ_i ≈ν_i, but should generalize to produce meaningful maps 𝒯_θ(c_new) 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.
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