# Uncoupled isotonic regression via minimum Wasserstein deconvolution

arXiv: https://arxiv.org/abs/1806.10648 · HTML: https://mlsanity.com/paper/1806.10648/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2018.

## Built on by (most cited first, 12 of 12)

- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Wasserstein K-means for clustering probability distributions](https://mlsanity.com/paper/2209.06975/) (2022)
- [Permuted and Unlinked Monotone Regression in ℝd: an approach based on mixture modeling and optimal transport](https://mlsanity.com/paper/2201.03528/) (2022)
- [Shuffled linear regression through graduated convex relaxation](https://mlsanity.com/paper/2209.15608/) (2022)
- [Generalized Talagrand Inequality for Sinkhorn Distance using Entropy Power Inequality](https://mlsanity.com/paper/2109.08430/) (2021)
- [Regularization for Shuffled Data Problems via Exponential Family Priors on the Permutation Group](https://mlsanity.com/paper/2111.01767/) (2021)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Bridged Clustering: Semi-Supervised Sparse Bridging](https://mlsanity.com/paper/2510.07182/) (2025)
- [Brenier Isotonic Regression](https://mlsanity.com/paper/2603.10452/) (2026)
- [Nonparametric mixture MLEs under Gaussian-smoothed optimal transport distance](https://mlsanity.com/paper/2112.02421/) (2021)
- [Statistical and computational challenges in ranking](https://mlsanity.com/paper/2512.21111/) (2025)
- [Finite-Sample Inference for Sparsely Permuted Linear Regression](https://mlsanity.com/paper/2601.14872/) (2026)

## Abstract (arXiv)

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 (xi, yi) where mathbbE[yi]=f(xi), i=1, ldots 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 x1, ldots, xn and y1, ldots, yn. In this work, we leverage tools from optimal transport theory to derive minimax rates under weak moments conditions on yi 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.