Constrained Denoising, Empirical Bayes, and Optimal Transport

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.