The Singularity Set of Optimal Transportation Maps

Optimal transportation plays an important role in many engineering fields, especially in deep learning. By the Brenier theorem, computing optimal transportation maps is reduced to solving Monge–Ampère equations, which in turn is equivalent to constructing Alexandrov polytopes. Furthermore, the regularity theory of Monge–Ampère equation explains mode collapsing issue in deep learning. Hence, computing and studying the singularity sets of OT maps become important. In this work, we generalize the concept of medial axis to power medial axis, which describes the singularity sets of optimal transportation maps. Then we propose a computational algorithm based on variational approach using power diagrams. Furthermore, we prove that when the measures are changed homotopically, the corresponding singularity sets of the optimal transportation maps are homotopy equivalent as well. Furthermore, we generalize the Fréchet distance concept and utilize the obliqueness condition to give a sufficient condition for the existence of singularities of optimal transportation maps between planar domains. The condition is formulated using the boundary curvature.