Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching

Denis Blessing, Lorenz Richter, Julius Berner, Egor Malitskiy, Gerhard Neumann
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:8496-8541, 2026.

Abstract

Sampling from unnormalized densities using diffusion models has emerged as a powerful paradigm. However, while recent approaches that use least-squares ‘matching’ objectives have improved scalability, they often necessitate significant trade-offs, such as restricting prior distributions or relying on unstable optimization schemes. By generalizing these methods as special forms of fixed-point iterations rooted in Nelson’s relation, we develop a new method that addresses these limitations. Our approach enables learning a stochastic transport map between arbitrary prior and target distributions with a single, scalable, and stable objective. Furthermore, we introduce a damped variant of this iteration that incorporates a regularization term to mitigate mode collapse. Empirically, we demonstrate that our method enables sampling at unprecedented scales while preserving mode diversity, achieving state-of-the-art results on complex synthetic densities and high-dimensional molecular benchmarks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-blessing26a, title = {Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching}, author = {Blessing, Denis and Richter, Lorenz and Berner, Julius and Malitskiy, Egor and Neumann, Gerhard}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {8496--8541}, 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/blessing26a/blessing26a.pdf}, url = {https://proceedings.mlr.press/v306/blessing26a.html}, abstract = {Sampling from unnormalized densities using diffusion models has emerged as a powerful paradigm. However, while recent approaches that use least-squares ‘matching’ objectives have improved scalability, they often necessitate significant trade-offs, such as restricting prior distributions or relying on unstable optimization schemes. By generalizing these methods as special forms of fixed-point iterations rooted in Nelson’s relation, we develop a new method that addresses these limitations. Our approach enables learning a stochastic transport map between arbitrary prior and target distributions with a single, scalable, and stable objective. Furthermore, we introduce a damped variant of this iteration that incorporates a regularization term to mitigate mode collapse. Empirically, we demonstrate that our method enables sampling at unprecedented scales while preserving mode diversity, achieving state-of-the-art results on complex synthetic densities and high-dimensional molecular benchmarks.} }
Endnote
%0 Conference Paper %T Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching %A Denis Blessing %A Lorenz Richter %A Julius Berner %A Egor Malitskiy %A Gerhard Neumann %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-blessing26a %I PMLR %P 8496--8541 %U https://proceedings.mlr.press/v306/blessing26a.html %V 306 %X Sampling from unnormalized densities using diffusion models has emerged as a powerful paradigm. However, while recent approaches that use least-squares ‘matching’ objectives have improved scalability, they often necessitate significant trade-offs, such as restricting prior distributions or relying on unstable optimization schemes. By generalizing these methods as special forms of fixed-point iterations rooted in Nelson’s relation, we develop a new method that addresses these limitations. Our approach enables learning a stochastic transport map between arbitrary prior and target distributions with a single, scalable, and stable objective. Furthermore, we introduce a damped variant of this iteration that incorporates a regularization term to mitigate mode collapse. Empirically, we demonstrate that our method enables sampling at unprecedented scales while preserving mode diversity, achieving state-of-the-art results on complex synthetic densities and high-dimensional molecular benchmarks.
APA
Blessing, D., Richter, L., Berner, J., Malitskiy, E. & Neumann, G.. (2026). Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:8496-8541 Available from https://proceedings.mlr.press/v306/blessing26a.html.

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