A new perspective on denoising based on optimal transport.

This article introduces a novel approach to the classical denoising problem—estimating an unobserved latent variable \(\varTheta\) from observations \(Z\) modeled probabilistically—by incorporating optimal transport (OT) theory. The authors define the OT-based denoiser as the estimator minimizing Bayes risk under a squared error loss while enforcing that its distribution matches the prior distribution \(G^*\) of \(\varTheta\), addressing limitations of the standard Bayes estimator which tends to over-shrink estimates and distort the latent distribution. They prove existence, uniqueness, and characterization of this OT-based denoiser via Monge's OT problem, and introduce two formulations: one using explicit knowledge of \(G^*\) and another relying only on the marginal distribution of observations and the likelihood model, leveraging Tweedie's formula for exponential families. Additionally, they propose a Kantorovich relaxation resembling multimarginal OT problems to establish existence of solutions and suggest computational strategies for finite data settings, including an empirical Bayes method to estimate the OT-based denoiser without directly estimating \(G^*\).