A Generative Sampler for distributions with possible discrete parameter based on Reversibility

A team of researchers led by Lei Li, Zhen Wang, and Lishuo Zhang has introduced a groundbreaking generative sampling framework that addresses a fundamental challenge in computational physics and machine learning: sampling from complex unnormalized distributions containing discrete or mixed variables. While existing methods like score-based and variational approaches work well for continuous domains, they struggle with discrete systems due to ill-defined gradients or high estimator variance. The new approach builds on the principle that detailed balance implies time-reversibility of equilibrium stochastic processes, enforcing this symmetry as a statistical constraint.

Specifically, the method uses a prescribed physical transition kernel (such as Metropolis-Hastings) and minimizes the Maximum Mean Discrepancy (MMD) between the joint distributions of forward and backward Markov trajectories. Crucially, this training procedure relies solely on energy evaluations via acceptance ratios, completely circumventing the need for target score functions or continuous relaxations that have limited previous approaches. This makes the framework universally applicable across diverse state spaces where traditional gradient-based methods fail.

The researchers demonstrated the versatility of their method on three distinct benchmarks: a continuous multi-modal Gaussian mixture, the discrete high-dimensional Ising model, and a challenging hybrid system coupling discrete indices with continuous dynamics. Experiments showed the framework accurately reproduces thermodynamic observables and captures mode-switching behavior across all regimes. The approach offers a physically grounded alternative for equilibrium sampling that could transform how researchers approach problems in statistical physics, materials science, and machine learning where discrete variables are essential.