Uncoupled isotonic regression via minimum Wasserstein deconvolution

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 (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.

Optimal Permutation Recovery in Permuted Monotone Matrix Model

“…For the univariate cases, Carpentier and Schlueter (2016) studied nonparametric estimation of smooth regression functions using a deconvolution approach. More recently, using the idea of optimal transport, a minimax optimal estimator of the underlying signals under the Wasserstein distance was obtained in Rigollet and Weed (2018) (see also Mao et al (2018) and references therein). However, these studies focus on the estimation of the underlying signal matrix Θ or ΘΠ, whereas the quantities of interest in the current study are the extreme columns.…”

On Estimation under Noisy Order Statistics

“…The goal is to estimate the correct permutation matrix and the linear parameter. Further extensions of [10] include [11] where the authors considered an additive noise regression model with the goal of estimating an unknown monotone increasing function of the input variables.…”

“…Before analyzing the performance of the suboptimal estimator given in (11), we state the next lemma (whose proof is provided in Appendix C), which offers a simplified expression for the estimator in (11).…”

“…In particular, we assess the performance of the two suboptimal estimators in (7) and in (11), as well as of the MLE in (17), by comparing them with the optimal estimator in Theorem 1.…”

Entropic optimal transport is maximum-likelihood deconvolution

“…As such, it has received significant attention from the statistics literature. More recently, it was shown that deconvolution has strong methodological and mathematical connections to optimal transport in the context of a problem known as uncoupled regression [RW18].…”