Uncoupled isotonic regression via minimum Wasserstein deconvolution

TL;DR

This paper introduces a novel approach to uncoupled isotonic regression using Wasserstein deconvolution, providing minimax rates and an efficient algorithm under weak moment conditions.

Contribution

It develops the first minimax optimal method for uncoupled isotonic regression using optimal transport techniques, with theoretical guarantees and practical algorithms.

Findings

Achieves minimax optimal rates for uncoupled isotonic regression.

Provides an efficient algorithm based on Wasserstein deconvolution.

Establishes bounds using moment-matching arguments relevant to mixture learning.

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 $\mathbb{E}[y_i]=f(x_i), i=1, \ldots n$. 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 $\{x_1, \ldots, x_n\}$ and $\{y_1, \ldots, y_n\}$. 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.