A New Perspective On Denoising Based On Optimal Transport

In the standard formulation of the denoising problem, one is given a probabilistic model relating a latent variable Theta ∈ Omega subset ℝm ; (m≥ 1) and an observation Z ∈ ℝd according to: Z mid Theta ~ p(·mid Theta) and Theta ~ 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 Theta 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, etc.). To rectify these drawbacks, 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, thanks to Tweedie's formula, when the likelihood model p(· mid θ) θ ∈ Omega 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.