# BELM: Bidirectional Explicit Linear Multi-step Sampler for Exact Inversion in Diffusion Models

arXiv: https://arxiv.org/abs/2410.07273 · HTML: https://mlsanity.com/paper/2410.07273/

## Key facts

- Cited by 29 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 61 of its 77 references that are in the map.
- First posted on arXiv 9 Oct 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 61)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [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)
- [Prompt-to-Prompt Image Editing with Cross Attention Control](https://mlsanity.com/paper/2208.01626/) (2022)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [ESRGAN: Enhanced Super-Resolution Generative Adversarial Networks](https://mlsanity.com/paper/1809.00219/) (2018)
- [Null-text Inversion for Editing Real Images using Guided Diffusion Models](https://mlsanity.com/paper/2211.09794/) (2022)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Zero-shot Image-to-Image Translation](https://mlsanity.com/paper/2302.03027/) (2023)
- [MM-LLMs: Recent Advances in MultiModal Large Language Models](https://mlsanity.com/paper/2401.13601/) (2024)
- [EDICT: Exact Diffusion Inversion via Coupled Transformations](https://mlsanity.com/paper/2211.12446/) (2022)

## Built on by (most cited first, 29 of 29)

- [Taming Rectified Flow for Inversion and Editing](https://mlsanity.com/paper/2411.04746/) (2024)
- [UniEdit-Flow: Unleashing Inversion and Editing in the Era of Flow Models](https://mlsanity.com/paper/2504.13109/) (2025)
- [Unveil Inversion and Invariance in Flow Transformer for Versatile Image Editing](https://mlsanity.com/paper/2411.15843/) (2024)
- [GenPO: Generative Diffusion Models Meet On-Policy Reinforcement Learning](https://mlsanity.com/paper/2505.18763/) (2025)
- [Mass Concept Erasure in Diffusion Models with Concept Hierarchy](https://mlsanity.com/paper/2601.03305/) (2026)
- [Adams Bashforth Moulton Solver for Inversion and Editing in Rectified Flow](https://mlsanity.com/paper/2503.16522/) (2025)
- [POLARIS: Projection-Orthogonal Least Squares for Robust and Adaptive Inversion in Diffusion Models](https://mlsanity.com/paper/2512.00369/) (2025)
- [FreeInv: Free Lunch for Improving DDIM Inversion](https://mlsanity.com/paper/2503.23035/) (2025)
- [ProEdit: Inversion-based Editing From Prompts Done Right](https://mlsanity.com/paper/2512.22118/) (2025)
- [FG-OrIU: Towards Better Forgetting via Feature-Gradient Orthogonality for Incremental Unlearning](https://mlsanity.com/paper/2601.13578/) (2026)
- [Efficiently Access Diffusion Fisher: Within the Outer Product Span Space](https://mlsanity.com/paper/2505.23264/) (2025)
- [Unleashing High-Quality Image Generation in Diffusion Sampling Using Second-Order Levenberg-Marquardt-Langevin](https://mlsanity.com/paper/2505.24222/) (2025)
- [On Efficiency-Effectiveness Trade-off of Diffusion-based Recommenders](https://mlsanity.com/paper/2510.17245/) (2025)
- [GenPO++: Generative Policy Optimization with Jacobian-free Likelihood Ratios](https://mlsanity.com/paper/2606.06967/) (2026)
- [Greed is Good: A Unifying Perspective on Guided Generation](https://mlsanity.com/paper/2502.08006/) (2025)
- [Learning Single Index Models with Diffusion Priors](https://mlsanity.com/paper/2505.21135/) (2025)
- [Instability in Diffusion ODEs: An Explanation for Inaccurate Image Reconstruction](https://mlsanity.com/paper/2506.18290/) (2025)
- [Contrastive Diffusion Alignment: Learning Structured Latents for Controllable Generation](https://mlsanity.com/paper/2510.14190/) (2025)
- [There and Back Again: On the relation between Noise and Image Inversions in Diffusion Models](https://mlsanity.com/paper/2410.23530/) (2024)
- [SPF-Portrait: Towards Pure Text-to-Portrait Customization with Semantic Pollution-Free Fine-Tuning](https://mlsanity.com/paper/2504.00396/) (2025)
- [BiFM: Bidirectional Flow Matching for Few-Step Image Editing and Generation](https://mlsanity.com/paper/2603.24942/) (2026)
- [Abstract Sound Fusion with Unconditional Inversion Models](https://mlsanity.com/paper/2506.11811/) (2025)
- [Morphology-optimized Multi-Scale Fusion: Combining Local Artifacts and Mesoscopic Semantics for Deepfake Detection and Localization](https://mlsanity.com/paper/2509.13776/) (2025)
- [An Iteration-Free Fixed-Point Estimator for Diffusion Inversion](https://mlsanity.com/paper/2512.08547/) (2025)
- [Stable and Near-Reversible Diffusion ODE Solvers for Image Editing](https://mlsanity.com/paper/2605.16399/) (2026)
- [Public Diffusion Models, Private Images: Key-Controlled Inversion for Conditional Reconstruction](https://mlsanity.com/paper/2606.22988/) (2026)
- [PortraitGen: Exemplar-Driven GRPO with Dual-Reward Guidance for Photorealistic Portrait Generation](https://mlsanity.com/paper/2606.26930/) (2026)
- [Multi-History-Step SDE Inversion for Image Editing with Superior Regional Awareness](https://mlsanity.com/paper/2609.06602/) (2026)
- [RAIN: Region-Aware Inversion Network for Semantic Watermark Extraction](https://mlsanity.com/paper/2609.14856/) (2026)

## Abstract (arXiv)

The inversion of diffusion model sampling, which aims to find the corresponding initial noise of a sample, plays a critical role in various tasks. Recently, several heuristic exact inversion samplers have been proposed to address the inexact inversion issue in a training-free manner. However, the theoretical properties of these heuristic samplers remain unknown and they often exhibit mediocre sampling quality. In this paper, we introduce a generic formulation, Bidirectional Explicit Linear Multi-step (BELM) samplers, of the exact inversion samplers, which includes all previously proposed heuristic exact inversion samplers as special cases. The BELM formulation is derived from the variable-stepsize-variable-formula linear multi-step method via integrating a bidirectional explicit constraint. We highlight this bidirectional explicit constraint is the key of mathematically exact inversion. We systematically investigate the Local Truncation Error (LTE) within the BELM framework and show that the existing heuristic designs of exact inversion samplers yield sub-optimal LTE. Consequently, we propose the Optimal BELM (O-BELM) sampler through the LTE minimization approach. We conduct additional analysis to substantiate the theoretical stability and global convergence property of the proposed optimal sampler. Comprehensive experiments demonstrate our O-BELM sampler establishes the exact inversion property while achieving high-quality sampling. Additional experiments in image editing and image interpolation highlight the extensive potential of applying O-BELM in varying applications.