Gauge Flow Matching for Efficient Constrained Generative Modeling over General Convex Set

Generative models, particularly diffusion and flow-matching approaches, have achieved remarkable success in various domains including image synthesis and robotic planning. However, a fundamental challenge remains: ensuring generated samples strictly satisfy problem-specific constraints—a crucial requirement for safety-critical applications and watermark embedding. Existing approaches, such as mirror maps and reflection methods, either support limited constraint sets or introduce significant computational overhead. In this paper, we develop gauge flow matching (GFM), a simple yet efficient framework for constrained generative modeling that introduces a bijective gauge mapping to transform generation over arbitrary compact convex sets into an equivalent process over the unit ball. Our GFM framework guarantees strict constraint satisfaction with low computational complexity and bounded distribution approximation errors. Extensive numerical experiments show that GFM outperforms existing methods in generation speed and quality across multiple benchmarks.