# Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling

arXiv: https://arxiv.org/abs/2412.20045 · HTML: https://mlsanity.com/paper/2412.20045/

## Key facts

- Cited by 5 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 7 of its 39 references that are in the map.
- First posted on arXiv 28 Dec 2024.

## Placed in

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

## Builds on (most cited first, 7 of 7)

- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Text-to-image Diffusion Models in Generative AI: A Survey](https://mlsanity.com/paper/2303.07909/) (2023)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [Monte Carlo guided Diffusion for Bayesian linear inverse problems](https://mlsanity.com/paper/2308.07983/) (2023)
- [Variational Bayesian Imaging with an Efficient Surrogate Score-based Prior](https://mlsanity.com/paper/2309.01949/) (2023)
- [Removing Structured Noise with Diffusion Models](https://mlsanity.com/paper/2302.05290/) (2023)

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

- [Conditional Mutual Information Based Diffusion Posterior Sampling for Solving Inverse Problems](https://mlsanity.com/paper/2501.02880/) (2025)
- [Coupled Data and Measurement Space Dynamics for Enhanced Diffusion Posterior Sampling](https://mlsanity.com/paper/2510.09676/) (2025)
- [Diffusion models for multivariate subsurface generation and efficient probabilistic inversion](https://mlsanity.com/paper/2507.15809/) (2025)
- [Energy-Tweedie: Score meets Score, Energy meets Energy](https://mlsanity.com/paper/2512.23818/) (2025)
- [Is your uncertainty map wrong, or is its target? Exact diagnostics for the Tweedie diagonal, and a gradient-free alternative](https://mlsanity.com/paper/2609.33786/) (2026)

## Abstract (arXiv)

Inverse problems exist in many disciplines of science and engineering. In computer vision, for example, tasks such as inpainting, deblurring, and super resolution can be effectively modeled as inverse problems. Recently, denoising diffusion probabilistic models (DDPMs) are shown to provide a promising solution to noisy linear inverse problems without the need for additional task specific training. Specifically, with the prior provided by DDPMs, one can sample from the posterior by approximating the likelihood. In the literature, approximations of the likelihood are often based on the mean of conditional densities of the reverse process, which can be obtained using Tweedie formula. To obtain a better approximation to the likelihood, in this paper we first derive a closed form formula for the covariance of the reverse process. Then, we propose a method based on finite difference method to approximate this covariance such that it can be readily obtained from the existing pretrained DDPMs, thereby not increasing the complexity compared to existing approaches. Finally, based on the mean and approximated covariance of the reverse process, we present a new approximation to the likelihood. We refer to this method as covariance-aware diffusion posterior sampling (CA-DPS). Experimental results show that CA-DPS significantly improves reconstruction performance without requiring hyperparameter tuning. The code for the paper is put in the supplementary materials.