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Isotonic regression is a standard problem in shape-constrained estimationwhere the goal is to estimate an unknown nondecreasing regression function $f$from independent pairs $(x_i, y_i)$ where $\mathbb{E}[y_i]=f(x_i), i=1, \ldotsn$. While this problem is well understood both statistically andcomputationally, much less is known about its uncoupled counterpart where oneis given only the unordered sets $\{x_1, \ldots, x_n\}$ and $\{y_1, \ldots,y_n\}$. In this work, we leverage tools from optimal transport theory to deriveminimax rates under weak moments conditions on $y_i$ and to give an efficientalgorithm achieving optimal rates. Both upper and lower bounds employmoment-matching arguments that are also pertinent to learning mixtures ofdistributions and deconvolution.

Uncoupled isotonic regression via minimum Wasserstein deconvolution