Improving Classifier-Free Guidance of Flow Matching via Manifold Projection

Jian-Feng Cai, Haixia Liu, Zhengyi Su, Chao Wang
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:10610-10635, 2026.

Abstract

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. Code is available at https://github.com/LeonSuZhengYi/CFG-MP.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-cai26g, title = {Improving Classifier-Free Guidance of Flow Matching via Manifold Projection}, author = {Cai, Jian-Feng and Liu, Haixia and Su, Zhengyi and Wang, Chao}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {10610--10635}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/cai26g/cai26g.pdf}, url = {https://proceedings.mlr.press/v306/cai26g.html}, abstract = {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. Code is available at https://github.com/LeonSuZhengYi/CFG-MP.} }
Endnote
%0 Conference Paper %T Improving Classifier-Free Guidance of Flow Matching via Manifold Projection %A Jian-Feng Cai %A Haixia Liu %A Zhengyi Su %A Chao Wang %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-cai26g %I PMLR %P 10610--10635 %U https://proceedings.mlr.press/v306/cai26g.html %V 306 %X 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. Code is available at https://github.com/LeonSuZhengYi/CFG-MP.
APA
Cai, J., Liu, H., Su, Z. & Wang, C.. (2026). Improving Classifier-Free Guidance of Flow Matching via Manifold Projection. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:10610-10635 Available from https://proceedings.mlr.press/v306/cai26g.html.

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