# Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching

arXiv: https://arxiv.org/abs/2405.18816 · HTML: https://mlsanity.com/paper/2405.18816/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 41 of its 68 references that are in the map.
- First posted on arXiv 29 May 2024.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 41)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [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)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [PULSE: Self-Supervised Photo Upsampling via Latent Space Exploration of Generative Models](https://mlsanity.com/paper/2003.03808/) (2020)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation](https://mlsanity.com/paper/2303.00848/) (2023)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (2016)
- [Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models](https://mlsanity.com/paper/2307.00619/) (2023)
- [Score-Based Diffusion Models as Principled Priors for Inverse Imaging](https://mlsanity.com/paper/2304.11751/) (2023)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Accelerator](https://mlsanity.com/paper/2405.07510/) (2024)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)

## Built on by (most cited first, 30 of 53)

- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling](https://mlsanity.com/paper/2504.10612/) (2025)
- [Flower: A Flow-Matching Solver for Inverse Problems](https://mlsanity.com/paper/2509.26287/) (2025)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans](https://mlsanity.com/paper/2501.16839/) (2025)
- [EdiVal-Agent: An Object-Centric Framework for Automated, Fine-Grained Evaluation of Multi-Turn Editing](https://mlsanity.com/paper/2509.13399/) (2025)
- [Denoising Score Distillation: From Noisy Diffusion Pretraining to One-Step High-Quality Generation](https://mlsanity.com/paper/2503.07578/) (2025)
- [The Generation Phases of Flow Matching: a Denoising Perspective](https://mlsanity.com/paper/2510.24830/) (2025)
- [Whitened Score Diffusion: A Structured Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2505.10311/) (2025)
- [Good Things Come in Pairs: Paired Autoencoders for Inverse Problems](https://mlsanity.com/paper/2505.06549/) (2025)
- [Unsupervised Imaging Inverse Problems with Diffusion Distribution Matching](https://mlsanity.com/paper/2506.14605/) (2025)
- [MAP Estimation with Denoisers: Convergence Rates and Guarantees](https://mlsanity.com/paper/2507.15397/) (2025)
- [Unlocking the Potential of Text-to-Image Diffusion with PAC-Bayesian Theory](https://mlsanity.com/paper/2411.17472/) (2024)
- [Multitask Learning with Stochastic Interpolants](https://mlsanity.com/paper/2508.04605/) (2025)
- [A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm](https://mlsanity.com/paper/2509.20511/) (2025)
- [Plug-and-Play Image Restoration with Flow Matching: A Continuous Viewpoint](https://mlsanity.com/paper/2512.04283/) (2025)
- [Aligning Latent Spaces with Flow Priors](https://mlsanity.com/paper/2506.05240/) (2025)
- [Flow Matching for Conditional MRI-CT and CBCT-CT Image Synthesis](https://mlsanity.com/paper/2510.04823/) (2025)
- [Latent Refinement via Flow Matching for Training-free Linear Inverse Problem Solving](https://mlsanity.com/paper/2511.06138/) (2025)
- [Image Restoration via Primal Dual Hybrid Gradient and Flow Generative Model](https://mlsanity.com/paper/2511.06748/) (2025)
- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)
- [Trajectory Stitching for Solving Inverse Problems with Flow-Based Models](https://mlsanity.com/paper/2602.08538/) (2026)
- [FMS2: Unified Flow Matching for Segmentation and Synthesis of Thin Structures](https://mlsanity.com/paper/2603.13659/) (2026)
- [Conditional flow matching for physics-constrained inverse problems with finite training data](https://mlsanity.com/paper/2603.14135/) (2026)
- [FluxFlow: Conservative Flow-Matching for Astronomical Image Super-Resolution](https://mlsanity.com/paper/2605.03749/) (2026)
- [NullFlow: One-Step Generative Reconstruction](https://mlsanity.com/paper/2606.22696/) (2026)
- [Moir'eXNet: Adaptive Multi-Scale Demoir'eing with Linear Attention Test-Time Training and Truncated Flow Matching Prior](https://mlsanity.com/paper/2506.15929/) (2025)
- [Restora-Flow: Mask-Guided Image Restoration with Flow Matching](https://mlsanity.com/paper/2511.20152/) (2025)
- [FlowSteer: Conditioning Flow Field for Consistent Image Restoration](https://mlsanity.com/paper/2512.08125/) (2025)
- [UPMRI: Unsupervised Parallel MRI Reconstruction via Projected Conditional Flow Matching](https://mlsanity.com/paper/2512.17493/) (2025)

## Abstract (arXiv)

Generative models based on flow matching have attracted significant attention for their simplicity and superior performance in high-resolution image synthesis. By leveraging the instantaneous change-of-variables formula, one can directly compute image likelihoods from a learned flow, making them enticing candidates as priors for downstream tasks such as inverse problems. In particular, a natural approach would be to incorporate such image probabilities in a maximum-a-posteriori (MAP) estimation problem. A major obstacle, however, lies in the slow computation of the log-likelihood, as it requires backpropagating through an ODE solver, which can be prohibitively slow for high-dimensional problems. In this work, we propose an iterative algorithm to approximate the MAP estimator efficiently to solve a variety of linear inverse problems. Our algorithm is mathematically justified by the observation that the MAP objective can be approximated by a sum of N ``local MAP'' objectives, where N is the number of function evaluations. By leveraging Tweedie's formula, we show that we can perform gradient steps to sequentially optimize these objectives. We validate our approach for various linear inverse problems, such as super-resolution, deblurring, inpainting, and compressed sensing, and demonstrate that we can outperform other methods based on flow matching. Code is available at https://github.com/YasminZhang/ICTM.