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.

翻译：在标准去噪问题表述中，给定一个概率模型，该模型关联潜在变量 $\Theta \in \Omega \subset \mathbb{R}^m \; (m\ge 1)$ 与观测 $Z \in \mathbb{R}^d$，满足：$Z \mid \Theta \sim p(\cdot\mid \Theta)$ 且 $\Theta \sim G^*$，目标是构造一个映射以从观测中恢复潜在变量。后验均值作为从 $Z$ 估计 $\Theta$ 的自然候选，能够在平方误差损失下达到最小贝叶斯风险，但代价是对 $Z$ 过度收缩，且一般情况下可能无法捕捉先验分布 $G^*$ 的几何特征（例如低维性、离散性、稀疏性等）。为解决这些缺陷，本文受最优传输理论的启发，对去噪问题提出新视角，并在总体水平上提出一种基于最优传输的新去噪器。我们严格证明，在模型的一般假设下，该基于最优传输的去噪器定义良好且唯一，并与蒙日最优传输问题的解紧密相关。进一步证明，在模型的适当可识别性假设下，通过求解一个在合适耦合空间上的线性松弛问题（类似于标准多边缘最优传输问题），该基于最优传输的去噪器可仅从 $Z$ 的边缘分布和模型的后验均值信息中恢复。特别地，得益于特威迪公式，当似然模型 $\{ p(\cdot \mid \theta) \}_{\theta \in \Omega}$ 为指数分布族时，该基于最优传输的去噪器可仅从 $Z$ 的边缘分布中恢复。总体上，我们这类最优传输型松弛问题本身具有研究价值，并为去噪问题提供了受计算最优传输丰富文献启发的替代数值方法。