# A New Perspective On Denoising Based On Optimal Transport

arXiv: https://arxiv.org/abs/2312.08135 · HTML: https://mlsanity.com/paper/2312.08135/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 80 references that are in the map.
- First posted on arXiv 13 Dec 2023.

## Builds on (most cited first, 20 of 20)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [A Theory of the Distortion-Perception Tradeoff in Wasserstein Space](https://mlsanity.com/paper/2107.02555/) (2021)
- [Multi-marginal optimal transport: theory and applications](https://mlsanity.com/paper/1406.0026/) (2014)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation](https://mlsanity.com/paper/1909.08733/) (2019)
- [Optimal Transport for Unsupervised Denoising Learning](https://mlsanity.com/paper/2108.02574/) (2021)
- [Optimal transport map estimation in general function spaces](https://mlsanity.com/paper/2212.03722/) (2022)
- [Two Modeling Strategies for Empirical Bayes Estimation](https://mlsanity.com/paper/1409.2677/) (2014)
- [Distribution-on-Distribution Regression via Optimal Transport Maps](https://mlsanity.com/paper/2104.09418/) (2021)
- [Empirical Bayes PCA in high dimensions](https://mlsanity.com/paper/2012.11676/) (2020)
- [Covariate powered cross-weighted multiple testing](https://mlsanity.com/paper/1701.05179/) (2017)
- [Invidious Comparisons: Ranking and Selection as Compound Decisions](https://mlsanity.com/paper/2012.12550/) (2020)
- [On Efficient and Scalable Computation of the Nonparametric Maximum Likelihood Estimator in Mixture Models](https://mlsanity.com/paper/2208.07514/) (2022)
- [Permuted and Unlinked Monotone Regression in ℝd: an approach based on mixture modeling and optimal transport](https://mlsanity.com/paper/2201.03528/) (2022)
- [Fully distribution-free center-outward rank tests for multiple-output regression and MANOVA](https://mlsanity.com/paper/2007.15496/) (2020)
- [Ground Truth Free Denoising by Optimal Transport](https://mlsanity.com/paper/2007.01575/) (2020)

## Built on by (most cited first, 2 of 2)

- [Distributional Shrinkage I: Universal Denoiser Beyond Tweedie's Formula](https://mlsanity.com/paper/2511.09500/) (2025)
- [Distributional Shrinkage II: Higher-Order Scores Encode Brenier Map](https://mlsanity.com/paper/2512.09295/) (2025)

## Abstract (arXiv)

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.