Uncoupled isotonic regression via minimum Wasserstein deconvolution

06/27/2018

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by   Philippe Rigollet, et al.

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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 {x_1, ..., x_n} and {y_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.