# Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems

arXiv: https://arxiv.org/abs/2303.05754 · HTML: https://mlsanity.com/paper/2303.05754/

## Key facts

- Cited by 132 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 27 of its 41 references that are in the map.
- First posted on arXiv 10 Mar 2023.

## Placed in

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

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [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)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction](https://mlsanity.com/paper/2112.05146/) (2021)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC](https://mlsanity.com/paper/2302.11552/) (2023)
- [End-to-End Variational Networks for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2004.06688/) (2020)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Solving 3D Inverse Problems using Pre-trained 2D Diffusion Models](https://mlsanity.com/paper/2211.10655/) (2022)
- [Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimation](https://mlsanity.com/paper/2106.05527/) (2021)
- [Bayesian MRI Reconstruction with Joint Uncertainty Estimation using Diffusion Models](https://mlsanity.com/paper/2202.01479/) (2022)
- [TorchRadon: Fast Differentiable Routines for Computed Tomography](https://mlsanity.com/paper/2009.14788/) (2020)
- [A Theoretical Justification for Image Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2302.01217/) (2023)
- [Sparse-view Cone Beam CT Reconstruction using Data-consistent Supervised and Adversarial Learning from Scarce Training Data](https://mlsanity.com/paper/2201.09318/) (2022)

## Built on by (most cited first, 30 of 132)

- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Manifold Preserving Guided Diffusion](https://mlsanity.com/paper/2311.16424/) (2023)
- [CFG++: Manifold-constrained Classifier Free Guidance for Diffusion Models](https://mlsanity.com/paper/2406.08070/) (2024)
- [Prompt-tuning latent diffusion models for inverse problems](https://mlsanity.com/paper/2310.01110/) (2023)
- [FlowDPS: Flow-Driven Posterior Sampling for Inverse Problems](https://mlsanity.com/paper/2503.08136/) (2025)
- [Direct Diffusion Bridge using Data Consistency for Inverse Problems](https://mlsanity.com/paper/2305.19809/) (2023)
- [3D MedDiffusion: A 3D Medical Latent Diffusion Model for Controllable and High-quality Medical Image Generation](https://mlsanity.com/paper/2412.13059/) (2024)
- [Regularization by Texts for Latent Diffusion Inverse Solvers](https://mlsanity.com/paper/2311.15658/) (2023)
- [Flower: A Flow-Matching Solver for Inverse Problems](https://mlsanity.com/paper/2509.26287/) (2025)
- [SITCOM: Step-wise Triple-Consistent Diffusion Sampling for Inverse Problems](https://mlsanity.com/paper/2410.04479/) (2024)
- [Deep Diffusion Image Prior for Efficient OOD Adaptation in 3D Inverse Problems](https://mlsanity.com/paper/2407.10641/) (2024)
- [ViBiDSampler: Enhancing Video Interpolation Using Bidirectional Diffusion Sampler](https://mlsanity.com/paper/2410.05651/) (2024)
- [Minority-Focused Text-to-Image Generation via Prompt Optimization](https://mlsanity.com/paper/2410.07838/) (2024)
- [Diffusion models for inverse problems](https://mlsanity.com/paper/2508.01975/) (2025)
- [Steerable Conditional Diffusion for Out-of-Distribution Adaptation in Medical Image Reconstruction](https://mlsanity.com/paper/2308.14409/) (2023)
- [SteerX: Creating Any Camera-Free 3D and 4D Scenes with Geometric Steering](https://mlsanity.com/paper/2503.12024/) (2025)
- [Implicit Image-to-Image Schrodinger Bridge for Image Restoration](https://mlsanity.com/paper/2403.06069/) (2024)
- [Solving Video Inverse Problems Using Image Diffusion Models](https://mlsanity.com/paper/2409.02574/) (2024)
- [Free Hunch: Denoiser Covariance Estimation for Diffusion Models Without Extra Costs](https://mlsanity.com/paper/2410.11149/) (2024)
- [DiffusionBlend: Learning 3D Image Prior through Position-aware Diffusion Score Blending for 3D Computed Tomography Reconstruction](https://mlsanity.com/paper/2406.10211/) (2024)
- [VISION-XL: High Definition Video Inverse Problem Solver using Latent Image Diffusion Models](https://mlsanity.com/paper/2412.00156/) (2024)
- [Geometry-Aware Attenuation Learning for Sparse-View CBCT Reconstruction](https://mlsanity.com/paper/2303.14739/) (2023)
- [SPIRiT-Diffusion: Self-Consistency Driven Diffusion Model for Accelerated MRI](https://mlsanity.com/paper/2304.05060/) (2023)
- [Inverse Problems with Diffusion Models: A MAP Estimation Perspective](https://mlsanity.com/paper/2407.20784/) (2024)
- [Likelihood-Scheduled Score-Based Generative Modeling for Fully 3D PET Image Reconstruction](https://mlsanity.com/paper/2412.04339/) (2024)
- [Patch-Based Diffusion Models Beat Whole-Image Models for Mismatched Distribution Inverse Problems](https://mlsanity.com/paper/2410.11730/) (2024)
- [Chain-of-Zoom: Extreme Super-Resolution via Scale Autoregression and Preference Alignment](https://mlsanity.com/paper/2505.18600/) (2025)
- [Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2505.20789/) (2025)
- [Amortized Posterior Sampling with Diffusion Prior Distillation](https://mlsanity.com/paper/2407.17907/) (2024)
- [Segmentation-guided MRI reconstruction for meaningfully diverse reconstructions](https://mlsanity.com/paper/2407.18026/) (2024)

## Abstract (arXiv)

Krylov subspace, which is generated by multiplying a given vector by the matrix of a linear transformation and its successive powers, has been extensively studied in classical optimization literature to design algorithms that converge quickly for large linear inverse problems. For example, the conjugate gradient method (CG), one of the most popular Krylov subspace methods, is based on the idea of minimizing the residual error in the Krylov subspace. However, with the recent advancement of high-performance diffusion solvers for inverse problems, it is not clear how classical wisdom can be synergistically combined with modern diffusion models. In this study, we propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if the tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG initialized with the denoised data ensures the data consistency update to remain in the tangent space. This negates the need to compute the manifold-constrained gradient (MCG), leading to a more efficient diffusion sampling method. Our method is applicable regardless of the parametrization and setting (i.e., VE, VP). Notably, we achieve state-of-the-art reconstruction quality on challenging real-world medical inverse imaging problems, including multi-coil MRI reconstruction and 3D CT reconstruction. Moreover, our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method. Code is available at https://github.com/HJ-harry/DDS