The empirical cost of optimal incomplete transportation

Abstract

We consider the problem of optimal incomplete transportation between the empirical measure on an i.i.d. uniform sample on the $d$-dimensional unit cube $[0,1]^{d}$ and the true measure. This is a family of problems lying in between classical optimal transportation and nearest neighbor problems. We show that the empirical cost of optimal incomplete transportation vanishes at rate $O_{P}(n^{-1/d})$, where $n$ denotes the sample size. In dimension $d\geq3$ the rate is the same as in classical optimal transportation, but in low dimension it is (much) higher than the classical rate.