Constrained Denoising, Empirical Bayes, and Optimal Transport

TL;DR

This paper introduces a constrained denoising framework combining optimal transport and empirical Bayes methods to improve denoising accuracy, with theoretical guarantees and practical applications in astronomy and baseball.

Contribution

It develops a modular approach to transform unconstrained empirical Bayes denoisers into constrained ones, providing explicit convergence rates and extending prior variance constraint results.

Findings

Derived explicit convergence rates for constrained denoising methods.

Applied the methodology to astronomy data on star abundances.

Used baseball data to assess batting skill improvements.

Abstract

In the statistical problem of denoising, Bayes and empirical Bayes methods can "overshrink" their output relative to the latent variables of interest. This work is focused on constrained denoising problems which mitigate such phenomena. At the oracle level, i.e., when the latent variable distribution is assumed known, we apply tools from the theory of optimal transport to characterize the solution to (i) variance-constrained, (ii) distribution-constrained, and (iii) general-constrained denoising problems. At the empirical level, i.e., when the latent variable distribution is not known, we use empirical Bayes methodology to estimate these oracle denoisers. Our approach is modular, and transforms any suitable (unconstrained) empirical Bayes denoiser into a constrained empirical Bayes denoiser. We prove explicit rates of convergence for our proposed methodologies, which both extend and sharpen existing asymptotic results that have previously considered only variance constraints. We apply our methodology in two applications: one in astronomy concerning the relative chemical abundances in a large catalog of red-clump stars, and one in baseball concerning minor- and major league batting skill for rookie players.