Uncoupled isotonic regression via minimum Wasserstein deconvolution

Abstract

Isotonic regression is a standard problem in shape-constrainedestimation where the goal is to estimate an unknown nondecreasingregression functionffrom independent pairs (xi,yi) whereE[yi] =f(xi),i= 1,...n. While this problem is well understood both statis-tically and computationally, much less is known about its uncoupledcounterpart where one is given only the unordered sets{x1,...,xn}and{y1,...,yn}. In this work, we leverage tools from optimal trans-port theory to derive minimax rates under weak moments conditionsonyiand to give an efficient algorithm achieving optimal rates. Bothupper and lower bounds employ moment-matching arguments that arealso pertinent to learning mixtures of distributions and deconvolution.