Markov chain Monte Carlo (MCMC) methods are simulated by local exploration of complex statistical distributions, and while bypassing the cumbersome requirement of a specific analytical expression for the target, this stochastic exploration of an uncertain parameter space comes at the expense of a large number of samples, and this computational complexity increases with parameter dimensionality. Although at the exploration level, some methods are proposed to accelerate the convergence of the algorithm, such as tempering, Hamiltonian Monte Carlo, Rao-redwellization, and scalable methods for better performance, it cannot avoid the stochastic nature of this exploration. We consider the target distribution as a mapping where the infinite-dimensional Eulerian space of the parameters consists of a number of deterministic submanifolds and propose a generalized energy metric, termed weighted Riesz energy, where a number of points is generated through pairwise interactions, to discretize rectifiable submanifolds. We study the properties of the point, called Riesz particle, and embed it into sequential MCMC, and we find that there will be higher acceptance rates with fewer evaluations, we validate it through experimental comparative analysis from a linear Gaussian state-space model with synthetic data and a non-linear stochastic volatility model with real-world data.

翻译：马尔可夫链蒙特卡洛（MCMC）方法通过局部探索复杂统计分布进行模拟，虽然绕过了对目标分布特定解析表达式的繁琐要求，但这种对不确定参数空间的随机探索以大量样本为代价，且计算复杂度随参数维度增加而增大。尽管在探索层面上，已有一些方法被提出以加速算法收敛，如温度调节、哈密顿蒙特卡洛、Rao-Blackwell化以及可扩展方法以提升性能，但仍无法避免这种探索的随机本质。我们将目标分布视为一种映射，其中参数的无限维欧拉空间由若干确定性子流形构成，并提出一种广义能量度量——加权Riesz能量——通过点之间的成对相互作用生成若干点，以实现对可整流子流形的离散化。我们研究了被称为Riesz粒子的点的性质，并将其嵌入到序贯MCMC中，发现这能以更少的评估次数实现更高的接受率。我们通过线性高斯状态空间模型（使用合成数据）和非线性随机波动模型（使用真实数据）的实验对比分析验证了这一结论。