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.

**Bio:**

Jonathan Weed is a graduate student at MIT, affiliated with the Department of Mathematics and the Statistics and Data Science Center. His research focuses on designing and analyzing robust procedures for learning from noisy information, especially when the underlying data has geometric structure. Recently, he has been particularly interested in statistical aspects of optimal transport.