Linearization of Monge-Amp\`ere equations and data science applications

Optimal transport has proven to have numerous applications in data science. In some of those, differentiating the transport map with respect to the densities in the Fr\'echet sense is required. In this work, we prove that for a compact, uniformly convex set Rᵈ and a C¹ curve of densities \pₜ\ C^1, () that are uniformly bounded away from zero and infinity, the curve of transport maps \ ₜ\ C^1, (), pushing Q forward to Pₜ (where Pₜ admits density pₜ and Q is sufficiently regular), is also C¹. The proof is based on applying the implicit function theorem to the Monge-Amp\`ere equation with natural boundary conditions. To achieve this, we establish the existence and uniqueness of solutions to the linearized Monge-Amp\`ere equation, an elliptic partial differential equation characterized by strictly oblique boundary conditions and a null zero-order term. We emphasize two significant applications of this result. Firstly, it enables non-parametric inference on the Monge map, paving the way for the construction of confidence bands for the transport map. Secondly, it establishes the regularity of the transport-based quantile regression function with respect to the covariates.