Effective Brenier Theorem

Abstract

Brenier's theorem is a landmark result in Optimal Transport. It postulates existence, monotonicity and uniqueness of an optimal map, with respect to the quadratic cost function, between two given probability measures (under some weak regularity conditions). We prove an effective version of Brenier's theorem: we show that for any two computable absolutely continuous measures on Rn, μ, and ν, with some restrictions on their support, there exists a computable convex function φ, whose gradient xφ is the optimal transport map between μ and ν. The main insight of the paper is the idea that an effective Brenier's theorem can be used to construct effective monotone maps on Rn with desired (non-)differentiability properties. We use it to solve several problems at the interface of algorithmic randomness and computable analysis. In particular, we show that z ∈ Rn is computably random if and only if every computable monotone function on Rn is differentiable at z. Furthermore, we prove the converse of the effective Aleksandrov theorem (Galicki 2015): we show that if z ∈ Rn is not computably random, there exists a computable convex function that is not twice differentiable at z. Finally, we prove several new characterisations of computable randomness in Rn: in terms of differentiability of computable measures, in terms of a particular Monge-Ampere equation and in terms of critical values of computable Lipschitz functions.