# Consistency Model is an Effective Posterior Sample Approximation for Diffusion Inverse Solvers

arXiv: https://arxiv.org/abs/2403.12063 · HTML: https://mlsanity.com/paper/2403.12063/

## Key facts

- Cited by 8 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 43 of its 66 references that are in the map.
- First posted on arXiv 9 Feb 2024.

## Placed in

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

## Builds on (most cited first, 30 of 43)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [BLIP: Bootstrapping Language-Image Pre-training for Unified Vision-Language Understanding and Generation](https://mlsanity.com/paper/2201.12086/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [CLIPScore: A Reference-free Evaluation Metric for Image Captioning](https://mlsanity.com/paper/2104.08718/) (2021)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [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)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Latent Consistency Models: Synthesizing High-Resolution Images with Few-Step Inference](https://mlsanity.com/paper/2310.04378/) (2023)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [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)
- [Universal Guidance for Diffusion Models](https://mlsanity.com/paper/2302.07121/) (2023)
- [PULSE: Self-Supervised Photo Upsampling via Latent Space Exploration of Generative Models](https://mlsanity.com/paper/2003.03808/) (2020)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [FreeDoM: Training-Free Energy-Guided Conditional Diffusion Model](https://mlsanity.com/paper/2303.09833/) (2023)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [Solving Inverse Problems with Latent Diffusion Models via Hard Data Consistency](https://mlsanity.com/paper/2307.08123/) (2023)
- [Certified Adversarial Robustness with Additive Noise](https://mlsanity.com/paper/1809.03113/) (2018)

## Built on by (most cited first, 8 of 8)

- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [LATINO-PRO: LAtent consisTency INverse sOlver with PRompt Optimization](https://mlsanity.com/paper/2503.12615/) (2025)
- [S4S: Solving for a Diffusion Model Solver](https://mlsanity.com/paper/2502.17423/) (2025)
- [Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2505.20789/) (2025)
- [Improving Diffusion-based Inverse Algorithms under Few-Step Constraint via Learnable Linear Extrapolation](https://mlsanity.com/paper/2503.10103/) (2025)
- [LVTINO: LAtent Video consisTency INverse sOlver for High Definition Video Restoration](https://mlsanity.com/paper/2510.01339/) (2025)
- [Audio Decoding by Inverse Problem Solving](https://mlsanity.com/paper/2409.07858/) (2024)
- [You Only Look One Step: Accelerating Backpropagation in Diffusion Sampling with Gradient Shortcuts](https://mlsanity.com/paper/2505.07477/) (2025)

## Abstract (arXiv)

Diffusion Inverse Solvers (DIS) are designed to sample from the conditional distribution pθ(X0|y), with a predefined diffusion model pθ(X0), an operator f(·), and a measurement y=f(x'0) derived from an unknown image x'0. Existing DIS estimate the conditional score function by evaluating f(·) with an approximated posterior sample drawn from pθ(X0|Xt). However, most prior approximations rely on the posterior means, which may not lie in the support of the image distribution, thereby potentially diverge from the appearance of genuine images. Such out-of-support samples may significantly degrade the performance of the operator f(·), particularly when it is a neural network. In this paper, we introduces a novel approach for posterior approximation that guarantees to generate valid samples within the support of the image distribution, and also enhances the compatibility with neural network-based operators f(·). We first demonstrate that the solution of the Probability Flow Ordinary Differential Equation (PF-ODE) with an initial value xt yields an effective posterior sample pθ(X0|Xt=xt). Based on this observation, we adopt the Consistency Model (CM), which is distilled from PF-ODE, for posterior sampling. Furthermore, we design a novel family of DIS using only CM. Through extensive experiments, we show that our proposed method for posterior sample approximation substantially enhance the effectiveness of DIS for neural network operators f(·) (e.g., in semantic segmentation). Additionally, our experiments demonstrate the effectiveness of the new CM-based inversion techniques. The source code is provided in the supplementary material.