Unstable optimal transport maps

Abstract: The stability of optimal transport maps with respect to perturbations of the marginals is a question of interest for several reasons, ranging from the justification of the linearized optimal transport framework to numerical analysis and statistics. Under various assumptions on the source measure, it is known that optimal transport maps are stable with respect to variations of the target measure. In this note, we focus on the mechanisms that can, on the contrary, lead to instability. We identify two of them, which we illustrate through examples of absolutely continuous source measures $\rho$ in $\mathbb{R}^d$ for which optimal transport maps are less stable, or even very unstable. We first show that instability may arise from the unboundedness of the density: we exhibit a source density on the unit ball of $\mathbb{R}^d$ which blows up superpolynomially at two points of the boundary and for which optimal transport maps are highly unstable. Then we prove that even for uniform densities on bounded open sets, optimal transport maps can be rather unstable close enough to configurations where uniqueness of optimal plans is lost.