The Singularity Set of Optimal Transportation Maps

Optimal transportation plays an important role in many engineering fields, especially in deep learning. By Brenier theorem, computating optimal transportation maps is reduced to solving Monge–Ampere equations, which in turn is equivalent to construct Alexandrov polytopes. Furthermore, the regularity theory of Monge–Ampere 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 homotopic equivalent as well.