A New Perspective On Denoising Based On Optimal Transport
This work offers a novel theoretical framework for denoising that could improve recovery of latent variables in statistical models, though it appears incremental as it builds on existing optimal transport theory.
The paper tackles the denoising problem by proposing an optimal transport-based denoiser to address over-shrinking and geometric feature loss in the posterior mean, proving its well-definedness and connection to Monge optimal transport problems under general assumptions.
In the standard formulation of the denoising problem, one is given a probabilistic model relating a latent variable $Θ\in Ω\subset \mathbb{R}^m \; (m\ge 1)$ and an observation $Z \in \mathbb{R}^d$ according to: $Z \mid Θ\sim p(\cdot\mid Θ)$ and $Θ\sim G^*$, and the goal is to construct a map to recover the latent variable from the observation. The posterior mean, a natural candidate for estimating $Θ$ 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 $\{ p(\cdot \mid θ) \}_{θ\in Ω}$ 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.