Uncoupled isotonic regression via minimum Wasserstein deconvolution

Abstract

Isotonic regression is a standard problem in shape-constrained estimation where the goal is to estimate an unknown nondecreasing regression function f from independent pairs (x_i, y_i) where E[y_i]=f(x_i), i=1, n. While this problem is well understood both statistically and computationally, much less is known about its uncoupled counterpart where one is given only the unordered sets _1, , x_n\ and _1, , y_n\. In this work, we leverage tools from optimal transport theory to derive minimax rates under weak moments conditions on y_i and to give an efficient algorithm achieving optimal rates. Both upper and lower bounds employ moment-matching arguments that are also pertinent to learning mixtures of distributions and deconvolution.

Tasks

Reproductions