Shape recognition via Wasserstein distance

Source abstract

The Kantorovich-Rubinstein-Wasserstein metric defines the distance between two probability measures μ \mu and ν \nu on R d + 1 {R^{d + 1}} by computing the cheapest way to transport the mass of μ \mu onto ν \nu , where the cost per unit mass transported is a given function c ( x , y ) c\left ( x, y \right ) on R 2 d + 2 {R^{2d + 2}} . Motivated by applications to shape recognition, we analyze this transportation problem with the cost c ( x , y ) = | x − y | 2 c\left ( x, y \right ) = {\left | {x - y} \right |^2} and measures supported on two curves in the plane, or more generally on the boundaries of two domains Ω , Λ ⊂ R d + 1 \Omega , \Lambda \subset {R^{d + 1}} . Unlike the theory for measures that are absolutely continuous with respect to Lebesgue, it turns out not to be the case that μ − a . e . x ∈ ∂ Ω \mu - a.e.x \in \partial \Omega is transported to a single image y ∈ ∂ Λ y \in \partial \Lambda ; however, we show that the images of x x are almost surely collinear and parallel the normal to ∂ Ω \partial \Omega at x x . If either domain is strictly convex, we deduce that the solution to the optimization problem is unique. When both domains are uniformly convex, we prove a regularity result showing that the images of x ∈ ∂ Ω x \in \partial \Omega are always collinear, and both images depend on x x in a continuous and (continuously) invertible way. This produces some unusual extremal doubly stochastic measures.