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：本文考虑从势函数支配的分布中进行采样的问题。我们提出了一种基于显式分数的确定性马尔可夫链蒙特卡洛方法，该方法导致粒子的确定性演化，而非随机微分方程演化。分数项通过正则化Wasserstein近似以闭式给出，其中使用通过采样近似的核卷积。我们在各种问题上展示了快速收敛性，并表明与未调整朗之万算法（ULA）和Metropolis调整朗之万算法（MALA）相比，在高斯分布情形下混合时间界的维数依赖性有所改善。此外，对于二次势函数，我们推导了每次迭代时分布的闭式表达式，刻画了方差缩减。实证结果表明，粒子以有序方式演化，分布在势函数的水平集等高线上。在贝叶斯逻辑回归背景下，所提方法的后验均值估计量比ULA和MALA更接近最大后验估计量。