# MAP Estimation with Denoisers: Convergence Rates and Guarantees

arXiv: https://arxiv.org/abs/2507.15397 · HTML: https://mlsanity.com/paper/2507.15397/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 46 references that are in the map.
- First posted on arXiv 21 Jul 2025.

## Builds on (most cited first, 27 of 27)

- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [Diffusion models as plug-and-play priors](https://mlsanity.com/paper/2206.09012/) (2022)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging](https://mlsanity.com/paper/2203.17061/) (2022)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation](https://mlsanity.com/paper/1512.07331/) (2015)
- [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024)
- [Scalable Plug-and-Play ADMM with Convergence Guarantees](https://mlsanity.com/paper/2006.03224/) (2020)
- [Provable Convergence of Plug-and-Play Priors with MMSE denoisers](https://mlsanity.com/paper/2005.07685/) (2020)
- [Convolutional Proximal Neural Networks and Plug-and-Play Algorithms](https://mlsanity.com/paper/2011.02281/) (2020)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Fixed-Point and Objective Convergence of Plug-and-Play Algorithms](https://mlsanity.com/paper/2104.10348/) (2021)
- [On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent](https://mlsanity.com/paper/2201.06133/) (2022)
- [Convergent regularization in inverse problems and linear plug-and-play denoisers](https://mlsanity.com/paper/2307.09441/) (2023)
- [Plug-and-play ISTA converges with kernel denoisers](https://mlsanity.com/paper/2004.03145/) (2020)

## Built on by (most cited first, 5 of 5)

- [When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis](https://mlsanity.com/paper/2509.24912/) (2025)
- [Landing with the Score: Riemannian Optimization through Denoising](https://mlsanity.com/paper/2509.23357/) (2025)
- [Beyond MMSE: Enhancing PnP Restoration with ProxiMAP](https://mlsanity.com/paper/2605.16396/) (2026)
- [Convergent Plug-and-Play Image Restoration with Annealed Noise Levels](https://mlsanity.com/paper/2609.32393/) (2026)
- [Principled MAP estimation for inverse problems: bridging the gap between convergence and performance](https://mlsanity.com/paper/2609.37529/) (2026)

## Abstract (arXiv)

Denoiser models have become powerful tools for inverse problems, enabling the use of pretrained networks to approximate the score of a smoothed prior distribution. These models are often used in heuristic iterative schemes aimed at solving Maximum a Posteriori (MAP) optimisation problems, where the proximal operator of the negative log-prior plays a central role. In practice, this operator is intractable, and practitioners plug in a pretrained denoiser as a surrogate-despite the lack of general theoretical justification for this substitution. In this work, we show that a simple algorithm, closely related to several used in practice, provably converges to the proximal operator under a log-concavity assumption on the prior p. We show that this algorithm can be interpreted as a gradient descent on smoothed proximal objectives. Our analysis thus provides a theoretical foundation for a class of empirically successful but previously heuristic methods.