A New Perspective On Denoising Based On Optimal Transport
Nicolas Garcia Trillos Bodhisattva Sen
In the standard formulation of the denoising problem, one is given a
probabilistic model relating a latent variable $\Theta \in \Omega \subset
\mathbb{R}^m \; (m\ge 1)$ and an observation $Z \in \mathbb{R}^d$ according to:
$Z \mid \Theta \sim p(\cdot\mid \Theta)$ and $\Theta \sim G^*$, and the goal is
to construct a map to recover the latent variable from the observation. The
posterior mean, a natural candidate for estimating $\Theta$ 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, in this paper we
take a new perspective on this denoising problem that is inspired by optimal
transport (OT) theory and use it to propose a new OT-based denoiser at the
population level setting. We rigorously prove that, under general assumptions
on the model, our OT-based denoiser is well-defined and unique, and is closely
connected to solutions to a Monge OT problem. We then prove that, under
appropriate identifiability assumptions on the model, our 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 a standard multimarginal
OT (MOT) problem. In particular, thanks to Tweedie's formula, when the
likelihood model $\{ p(\cdot \mid \theta) \}_{\theta \in \Omega}$ 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.