Unstable optimal transport maps
Résumé
The stability of optimal transport maps with respect to perturbations of the marginals is a question of interest for several reasons, ranging from numerical analysis and statistics to the justification of the linearized optimal transport framework. 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. 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 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 sufficiently close to configurations where uniqueness of optimal plans is lost.
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