We consider the problem of sampling from a distribution governed by a potential function. This work proposes an explicit score-based MCMC method that is deterministic, resulting in a deterministic evolution for particles rather than a stochastic differential equation evolution. The score term is given in closed form by a regularized Wasserstein proximal, using a kernel convolution that is approximated by sampling. We demonstrate fast convergence on various problems and show improved dimensional dependence of mixing time bounds for the case of Gaussian distributions compared to the unadjusted Langevin algorithm (ULA) and the Metropolis-adjusted Langevin algorithm (MALA). We additionally derive closed form expressions for the distributions at each iterate for quadratic potential functions, characterizing the variance reduction. Empirical results demonstrate that the particles behave in an organized manner, lying on level set contours of the potential. Moreover, the posterior mean estimator of the proposed method is shown to be closer to the maximum a-posteriori estimator compared to ULA and MALA, in the context of Bayesian logistic regression.

翻译：我们考虑从势函数控制下的分布中进行采样的问题。本文提出了一种基于显式分数匹配的确定性马尔可夫链蒙特卡洛（MCMC）方法，该方法为粒子提供确定性演化过程，而非随机微分方程演化机制。分数项通过正则化Wasserstein近似的闭式表达式给出，其中涉及通过采样近似的核卷积操作。我们通过多种问题验证了算法的快速收敛性，并证明在高斯分布情形下，其混合时间界限对维度的依赖优于未调整的朗之万算法（ULA）和Metropolis调整的朗之万算法（MALA）。针对二次势函数，我们进一步推导出每次迭代分布密度的闭式表达式，并刻画了方差的缩减特性。实验结果表明，粒子在势函数水平集曲线上呈现有序分布形态。在贝叶斯逻辑回归场景中，所提方法的后验均值估计更接近最大后验估计，优于ULA和MALA。