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

TL;DR

This work proposes a unified, target-gradient-free generative sampling framework applicable across diverse state spaces, offering a physically grounded and universally applicable alternative for equilibrium sampling.

Abstract

Learning to sample from complex unnormalized distributions is a fundamental challenge in computational physics and machine learning. While score-based and variational methods have achieved success in continuous domains, extending them to discrete or mixed-variable systems remains difficult due to ill-defined gradients or high variance in estimators. We propose a unified, target-gradient-free generative sampling framework applicable across diverse state spaces. Building on the fact that detailed balance implies the time-reversibility of the equilibrium stochastic process, we enforce this symmetry as a statistical constraint. Specifically, using a prescribed physical transition kernel (such as Metropolis-Hastings), we minimize 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, circumventing the need for target score functions or continuous relaxations. We demonstrate the versatility of our method on three distinct benchmarks: (1) a continuous multi-modal Gaussian mixture, (2) the discrete high-dimensional Ising model, and (3) a challenging hybrid system coupling discrete indices with continuous dynamics. Experiments show that our framework accurately reproduces thermodynamic observables and captures mode-switching behavior across all regimes, offering a physically grounded and universally applicable alternative for equilibrium sampling.

Figures (5)

• Figure 1: Qualitative results on the 2D Gaussian mixture. Top-left: Ground truth target density. Top-right: Learned density by our model. Bottom row: Marginal distributions along the first (left) and second (right) dimensions. The learned density accurately captures both modes and their relative weights.
• Figure 2: Training curves
• Figure 3: Visual evaluation of the generative fidelity on the balanced double-well system. (a) Conditional distributions of the continuous coordinate $x$ given the discrete mode $k$. The generated samples perfectly align with the theoretical probability density functions across all three modes ($\mu \in \{1.0, 9.0, 25.0\}$), demonstrating the model's precise capture of local well geometries and thermal fluctuations. (b) The marginal distribution of $x$ aggregated over all modes. The symmetric and globally matched profile confirms the absence of mode collapse and validates the balanced exploration of the phase space.
• Figure 4: Generative fidelity for the 2D Ising Model in the high-temperature phase ($L=3, \beta=0.2$). (Top-left) Magnetization distribution showing a disordered state centered at zero. (Top-right) Energy distribution matching the theoretical spectrum. (Bottom) Probabilities of the Top-100 most likely configurations, demonstrating exact alignment with analytical enumeration.
• Figure 5: Generative fidelity for the 2D Ising Model in the low-temperature phase ($L=3, \beta=0.5$). (Top-left) Magnetization distribution capturing the broadened profile indicative of ordered states. (Top-right) Energy distribution properly shifted to lower states. (Bottom) Top-100 configuration probabilities, maintaining alignment despite stronger spin correlations.

Theorems & Definitions (9)

• Definition 2.1: Reversible Stochastic Process
• Definition 2.2: Detailed Balancebinder1992monte
• Proposition 2.1: Stationarity and Symmetryswain1984handbook
• Remark 3.1: Framework Extensibility
• Lemma 4.1
• Lemma 4.2
• proof
• Theorem 4.1
• Proposition 4.1