Optimal transportation with an oscillation-type cost: the one-dimensional case

TL;DR

This paper proves the existence of an optimal transport map minimizing a cost based on the maximal oscillation at a scale, with applications in privacy-respectful data transmission, but only in one dimension.

Contribution

It introduces a new oscillation-based cost function for optimal transport and proves existence of solutions in the one-dimensional case.

Findings

Existence of an optimal transport map for the oscillation cost in 1D

Application potential in privacy-respectful data transmission

Method relies on monotonicity considerations

Abstract

The main result of this paper is the existence of an optimal transport map $T$ between two given measures $\mu$ and $\nu$, for a cost which considers the maximal oscillation of $T$ at scale $\delta$, given by $\omega_\delta(T):=\sup_{|x-y|<\delta}|T(x)-T(y)|$. The minimization of this criterion finds applications in the field of privacy-respectful data transmission. The existence proof unfortunately only works in dimension one and is based on some monotonicity considerations.