In the statistical problem of denoising, it is well-known that Bayes and empirical Bayes methods lead to denoisers which ``overshrink'' their output relative to the latent variables of interest; this work is focused on constrained denoising problems (corresponding to variance constraints, distribution constraints, or more general types of constraints) which preclude such phenomena. More precisely, we leverage recent developments in optimal transport to provide a general empirical Bayes framework for estimating the optimal constrained denoiser. We provide various asymptotic and non-asymptotic guarantees for the convergence of this empirical Bayes procedure, and we explore some applications to simulated and real data.
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