# Improving Classifier-Free Guidance of Flow Matching via Manifold Projection

arXiv: https://arxiv.org/abs/2601.21892 · HTML: https://mlsanity.com/paper/2601.21892/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 30 of its 40 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [CLIPScore: A Reference-free Evaluation Metric for Image Captioning](https://mlsanity.com/paper/2104.08718/) (2021)
- [ImageReward: Learning and Evaluating Human Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2304.05977/) (2023)
- [FLUX.1 Kontext: Flow Matching for In-Context Image Generation and Editing in Latent Space](https://mlsanity.com/paper/2506.15742/) (2025)
- [Pick-a-Pic: An Open Dataset of User Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2305.01569/) (2023)
- [GenEval: An Object-Focused Framework for Evaluating Text-to-Image Alignment](https://mlsanity.com/paper/2310.11513/) (2023)
- [Human Preference Score v2: A Solid Benchmark for Evaluating Human Preferences of Text-to-Image Synthesis](https://mlsanity.com/paper/2306.09341/) (2023)
- [On Distillation of Guided Diffusion Models](https://mlsanity.com/paper/2210.03142/) (2022)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [CFG++: Manifold-constrained Classifier Free Guidance for Diffusion Models](https://mlsanity.com/paper/2406.08070/) (2024)
- [Eliminating Oversaturation and Artifacts of High Guidance Scales in Diffusion Models](https://mlsanity.com/paper/2410.02416/) (2024)
- [CFG-Zero*: Improved Classifier-Free Guidance for Flow Matching Models](https://mlsanity.com/paper/2503.18886/) (2025)
- [Zigzag Diffusion Sampling: Diffusion Models Can Self-Improve via Self-Reflection](https://mlsanity.com/paper/2412.10891/) (2024)
- [Diff2Flow: Training Flow Matching Models via Diffusion Model Alignment](https://mlsanity.com/paper/2506.02221/) (2025)
- [Convergence Rate of Incremental Gradient and Newton Methods](https://mlsanity.com/paper/1510.08562/) (2015)
- [Characteristic Guidance: Non-linear Correction for Diffusion Model at Large Guidance Scale](https://mlsanity.com/paper/2312.07586/) (2023)
- [Interpreting and Improving Diffusion Models from an Optimization Perspective](https://mlsanity.com/paper/2306.04848/) (2023)
- [Rectified-CFG++ for Flow Based Models](https://mlsanity.com/paper/2510.07631/) (2025)
- [Single Loop Gaussian Homotopy Method for Non-convex Optimization](https://mlsanity.com/paper/2203.05717/) (2022)
- [Continuation Path Learning for Homotopy Optimization](https://mlsanity.com/paper/2307.12551/) (2023)
- [Towards a Golden Classifier-Free Guidance Path via Foresight Fixed Point Iterations](https://mlsanity.com/paper/2510.21512/) (2025)

## Built on by (most cited first, 3 of 3)

- [PathGuide: Dynamic Classifier-Free Guidance via On-Policy Transport Alignment](https://mlsanity.com/paper/2608.29107/) (2026)
- [Classifier-Free Guidance in Flow Matching: Non-Autonomous Potentials, Overshoot, and Posterior-Mean Control](https://mlsanity.com/paper/2609.24287/) (2026)
- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)

## Abstract (arXiv)

Classifier-free guidance (CFG) is a widely used technique for controllable generation in diffusion and flow-based models. Despite its empirical success, CFG relies on a heuristic linear extrapolation that is often sensitive to the guidance scale. In this work, we provide a principled interpretation of CFG through the lens of optimization. We demonstrate that the velocity field in flow matching corresponds to the gradient of a sequence of smoothed distance functions, which guides latent variables toward the scaled target image set. This perspective reveals that the standard CFG formulation is an approximation of this gradient, where the prediction gap, the discrepancy between conditional and unconditional outputs, governs guidance sensitivity. Leveraging this insight, we reformulate the CFG sampling as a homotopy optimization with a manifold constraint. This formulation necessitates a manifold projection step, which we implement via an incremental gradient descent scheme during sampling. To improve computational efficiency and stability, we further enhance this iterative process with Anderson Acceleration without requiring additional model evaluations. Our proposed methods are training-free and consistently refine generation fidelity, prompt alignment, and robustness to the guidance scale. We validate their effectiveness across diverse benchmarks, demonstrating significant improvements on large-scale models such as DiT-XL-2-256, Flux, and Stable Diffusion 3.5.