# Principled MAP estimation for inverse problems: bridging the gap between convergence and performance

arXiv: https://arxiv.org/abs/2609.37529 · HTML: https://mlsanity.com/paper/2609.37529/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 42 references that are in the map.
- First posted on arXiv 29 Sep 2026.

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [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)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024)
- [Flower: A Flow-Matching Solver for Inverse Problems](https://mlsanity.com/paper/2509.26287/) (2025)
- [Plug-and-Play image restoration with Stochastic deNOising REgularization](https://mlsanity.com/paper/2402.01779/) (2024)
- [Normalizing constants of log-concave densities](https://mlsanity.com/paper/1707.00460/) (2017)
- [MAP Estimation with Denoisers: Convergence Rates and Guarantees](https://mlsanity.com/paper/2507.15397/) (2025)
- [Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution](https://mlsanity.com/paper/2104.11987/) (2021)
- [Beyond MMSE: Enhancing PnP Restoration with ProxiMAP](https://mlsanity.com/paper/2605.16396/) (2026)

## Abstract (arXiv)

Pretrained denoisers provide a powerful way to incorporate image priors into restoration algorithms. Plug-and-Play and RED approaches exploit fixed-noise-level denoisers within first-order optimization schemes, with convergence guarantees, but often struggle to achieve high-quality reconstruction on severely ill-posed inverse problems. In contrast, recent state-of-the-art approaches leverage denoisers derived from flow- or diffusion-based generative models and evaluate them along a sequence of decreasing noise levels. While these methods achieve strong empirical performance, their convergence theory remains limited. In this paper, we bridge this gap by specifically designing an algorithm that combines denoisers at decreasing noise levels with a schedule tailored to ensure convergence. From a Bayesian perspective, we prove that our method converges to a textitMaximum a Posteriori (MAP) estimate, under suitable assumptions. Subsequently, we apply our method to various ill-posed inverse problems and show that it surpasses convergent methods while competing with state-of-the-art empirical ones.