A fast approach to optimal transport: the back-and-forth method

Matt Jacobs, Flavien Léger

Abstract: We present a method to efficiently solve the optimal transportation problem for a general class of strictly convex costs. Given two probability measures supported on a discrete grid with n points we compute the optimal transport map between them in $$O(n\log (n))$$ O ( n log ( n ) ) operations and O(n) storage space. Our approach allows us to solve optimal transportation problems on spatial grids as large as $$4096\times 4096$$ 4096 × 4096 and $$384\times 384\times 384$$ 384 × 384 × 384 in a matter of minutes.