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

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.

In many areas of science and machine learning, the goal is to generate new data that fits a particular task — for example, plausible shapes a molecule might fold into, or configurations of a physical system. Usually, we train generative models for this by showing them a large dataset of real examples to imitate. But in many important cases no such dataset exists. Instead, all we have is a feedback signal: a formula that, given any candidate configuration, tells us how good or how likely it is, without ever providing examples to learn from. We study how to train generative models in exactly this setting — learning to produce diverse, realistic samples using only this feedback signal, with no access to data. Our method, the Bridge Matching Sampler, works through a simple repeated loop: it runs the current model, connects its outputs to fresh starting points with simple "bridge" paths, and trains the model to imitate the average of those bridges. We further add a "damping" adjustment that steadies training and stops the model from neglecting less common outcomes, allowing it to scale to much harder, higher-dimensional problems than before.