# Inverse Problems with Diffusion Models: A MAP Estimation Perspective

arXiv: https://arxiv.org/abs/2407.20784 · HTML: https://mlsanity.com/paper/2407.20784/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 28 of its 36 references that are in the map.
- First posted on arXiv 27 Jul 2024.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 28 of 28)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Context Encoders: Feature Learning by Inpainting](https://mlsanity.com/paper/1604.07379/) (2016)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [Semantic Image Inpainting with Deep Generative Models](https://mlsanity.com/paper/1607.07539/) (2016)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems](https://mlsanity.com/paper/2303.05754/) (2023)
- [DMPlug: A Plug-in Method for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2405.16749/) (2024)
- [3DDesigner: Towards Photorealistic 3D Object Generation and Editing with Text-guided Diffusion Models](https://mlsanity.com/paper/2211.14108/) (2022)
- [Improving Diffusion Models for Inverse Problems Using Optimal Posterior Covariance](https://mlsanity.com/paper/2402.02149/) (2024)

## Built on by (most cited first, 9 of 9)

- [Rethinking Diffusion Posterior Sampling: From Conditional Score Estimator to Maximizing a Posterior](https://mlsanity.com/paper/2501.18913/) (2025)
- [Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments](https://mlsanity.com/paper/2506.00214/) (2025)
- [HybridLinker: Topology-Guided Posterior Sampling for Enhanced Diversity and Validity in 3D Molecular Linker Generation](https://mlsanity.com/paper/2502.17349/) (2025)
- [Local MAP Sampling for Diffusion Models](https://mlsanity.com/paper/2510.07343/) (2025)
- [Image Restoration via Diffusion Models with Dynamic Resolution](https://mlsanity.com/paper/2605.14267/) (2026)
- [Beyond MMSE: Enhancing PnP Restoration with ProxiMAP](https://mlsanity.com/paper/2605.16396/) (2026)
- [Iterative CT Reconstruction via Latent Variable Optimization of Shallow Diffusion Models](https://mlsanity.com/paper/2408.03156/) (2024)
- [Inverse problems with diffusion models: MAP estimation via mode-seeking loss](https://mlsanity.com/paper/2512.10524/) (2025)
- [GenDA: Generative Data Assimilation on Complex Urban Areas via Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2601.11440/) (2026)

## Abstract (arXiv)

Inverse problems have many applications in science and engineering. In Computer vision, several image restoration tasks such as inpainting, deblurring, and super-resolution can be formally modeled as inverse problems. Recently, methods have been developed for solving inverse problems that only leverage a pre-trained unconditional diffusion model and do not require additional task-specific training. In such methods, however, the inherent intractability of determining the conditional score function during the reverse diffusion process poses a real challenge, leaving the methods to settle with an approximation instead, which affects their performance in practice. Here, we propose a MAP estimation framework to model the reverse conditional generation process of a continuous time diffusion model as an optimization process of the underlying MAP objective, whose gradient term is tractable. In theory, the proposed framework can be applied to solve general inverse problems using gradient-based optimization methods. However, given the highly non-convex nature of the loss objective, finding a perfect gradient-based optimization algorithm can be quite challenging, nevertheless, our framework offers several potential research directions. We use our proposed formulation to develop empirically effective algorithms for image restoration. We validate our proposed algorithms with extensive experiments over multiple datasets across several restoration tasks.