# Flow Matching Posterior Sampling: A Training-free Conditional Generation for Flow Matching

arXiv: https://arxiv.org/abs/2411.07625 · HTML: https://mlsanity.com/paper/2411.07625/

## Key facts

- Cited by 4 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 34 of its 50 references that are in the map.
- First posted on arXiv 12 Nov 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 34)

- [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)
- [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)
- [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)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generation](https://mlsanity.com/paper/2208.12242/) (2022)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Magic3D: High-Resolution Text-to-3D Content Creation](https://mlsanity.com/paper/2211.10440/) (2022)
- [Reproducible scaling laws for contrastive language-image learning](https://mlsanity.com/paper/2212.07143/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [FreeDoM: Training-Free Energy-Guided Conditional Diffusion Model](https://mlsanity.com/paper/2303.09833/) (2023)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [Manifold Preserving Guided Diffusion](https://mlsanity.com/paper/2311.16424/) (2023)
- [ControlNet++: Improving Conditional Controls with Efficient Consistency Feedback](https://mlsanity.com/paper/2404.07987/) (2024)
- [Contrastive Energy Prediction for Exact Energy-Guided Diffusion Sampling in Offline Reinforcement Learning](https://mlsanity.com/paper/2304.12824/) (2023)
- [Implicit Style-Content Separation using B-LoRA](https://mlsanity.com/paper/2403.14572/) (2024)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [TFG: Unified Training-Free Guidance for Diffusion Models](https://mlsanity.com/paper/2409.15761/) (2024)
- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)
- [Guidance with Spherical Gaussian Constraint for Conditional Diffusion](https://mlsanity.com/paper/2402.03201/) (2024)
- [Flow Matching with Gaussian Process Priors for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2410.03024/) (2024)

## Built on by (most cited first, 4 of 4)

- [TFTF: Training-Free Targeted Flow for Conditional Sampling](https://mlsanity.com/paper/2602.12932/) (2026)
- [Guiding Diffusion with Deep Geometric Moments: Balancing Fidelity and Variation](https://mlsanity.com/paper/2505.12486/) (2025)
- [Getting Your Guidance Weights Right in diffusion and flow-matching posterior sampling](https://mlsanity.com/paper/2610.03503/) (2026)
- [BridgeTwist: Twisting Schr"odinger Bridges for Training-Free Conditional Sampling](https://mlsanity.com/paper/2610.18658/) (2026)

## Abstract (arXiv)

Training-free conditional generation based on flow matching aims to leverage pre-trained unconditional flow matching models to perform conditional generation without retraining. Recently, a successful training-free conditional generation approach incorporates conditions via posterior sampling, which relies on the availability of a score function in the unconditional diffusion model. However, flow matching models do not possess an explicit score function, rendering such a strategy inapplicable. Approximate posterior sampling for flow matching has been explored, but it is limited to linear inverse problems. In this paper, we propose Flow Matching-based Posterior Sampling (FMPS) to expand its application scope. We introduce a correction term by steering the velocity field. This correction term can be reformulated to incorporate a surrogate score function, thereby bridging the gap between flow matching models and score-based posterior sampling. Hence, FMPS enables the posterior sampling to be adjusted within the flow matching framework. Further, we propose two practical implementations of the correction mechanism: one aimed at improving generation quality, and the other focused on computational efficiency. Experimental results on diverse conditional generation tasks demonstrate that our method achieves superior generation quality compared to existing state-of-the-art approaches, validating the effectiveness and generality of FMPS.