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

2026-02-28

AI Summary

This work addresses the challenges of mode collapse and training instability commonly encountered in sampling from high-dimensional, complex unnormalized densities. The authors propose a generalized fixed-point diffusion matching framework that unifies the diffusion process as a learnable stochastic transport map grounded in Nelson’s relations. By incorporating damping iterations and regularization mechanisms, the method establishes a single-objective, scalable optimization strategy that eliminates restrictive assumptions on prior distributions. Experimental results on both synthetic densities and high-dimensional molecular data demonstrate that the approach significantly enhances sampling stability and diversity, achieving state-of-the-art performance. Moreover, it enables unprecedented scalability while effectively preserving multimodal structures inherent in the target distributions.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

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, called Bridge Matching Sampler (BMS). 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 and further stabilize training. 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.

Problem

• diffusion models
• unnormalized densities
• scalable sampling
• mode collapse
• fixed-point iterations

Innovation

• diffusion matching
• fixed-point iteration
• stochastic transport map
• mode collapse mitigation
• scalable sampling