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Abstract In the standard formulation of the classical denoising problem, one is given a probabilistic model relating a latent variable $$\varTheta \in \varOmega \subset{\mathbb{R}}^{m} \; (m\ge 1)$$ and an observation $$Z \in{\mathbb{R}}^{d}$$ according to $$Z \mid \varTheta \sim p(\cdot \mid \varTheta )$$ and $$\varTheta \sim G^{*}$$, and the goal is to construct a map to recover the latent variable from the observation. The posterior mean, a natural candidate for estimating $$\varTheta $$ from $$Z$$, attains the minimum Bayes risk (under the squared error loss) but at the expense of over-shrinking the $$Z$$, and in general may fail to capture the geometric features of the prior distribution $$G^{*}$$ (e.g. low dimensionality, discreteness, sparsity). To rectify these drawbacks, in this paper we take a new perspective on this denoising problem that is inspired by optimal transport (OT) theory and use it to study a different, OT-based, denoiser at the population level setting. We rigorously prove that, under general assumptions on the model, this OT-based denoiser is mathematically well-defined and unique, and is closely connected to the solution to a Monge OT problem. We then prove that, under appropriate identifiability assumptions on the model, the OT-based denoiser can be recovered solely from information of the marginal distribution of $$Z$$ and the posterior mean of the model, after solving a linear relaxation problem over a suitable space of couplings that is reminiscent of standard multimarginal OT problems. In particular, due to Tweedie’s formula, when the likelihood model $$\{ p(\cdot \mid \theta ) \}_{\theta \in \varOmega }$$ is an exponential family of distributions, the OT-based denoiser can be recovered solely from the marginal distribution of $$Z$$. In general, our family of OT-like relaxations is of interest in its own right and for the denoising problem suggests alternative numerical methods inspired by the rich literature on computational OT.