A new particle-based sampling and approximate inference method, based on
electrostatics and Newton mechanics principles, is introduced with theoretical
ground, algorithm design and experimental validation. This method simulates an
interacting particle system (IPS) where particles, i.e. the freely-moving
negative charges and spatially-fixed positive charges with magnitudes
proportional to the target distribution, interact with each other via
attraction and repulsion induced by the resulting electric fields described by
Poisson's equation. The IPS evolves towards a steady-state where the
distribution of negative charges conforms to the target distribution. This
physics-inspired method offers deterministic, gradient-free sampling and
inference, achieving comparable performance as other particle-based and MCMC
methods in benchmark tasks of inferring complex densities, Bayesian logistic
regression and dynamical system identification. A discrete-time, discrete-space
algorithmic design, readily extendable to continuous time and space, is
provided for usage in more general inference problems occurring in
probabilistic machine learning scenarios such as Bayesian inference, generative
modelling, and beyond.

基于粒子的采样和近似推断方法，基于静电学和牛顿力学原理，提出了一种新的方法，通过模拟相互作用的粒子系统来实现确定性、无梯度的采样和推断，可应用于概率机器学习中的贝叶斯推断、生成模型等问题。

基于静电的粒子采样与近似推断

Electrostatics-based particle sampling and approximate inference

We study an interacting particle system in $\mathbf{R}^d$ motivated by Stein
variational gradient descent [Q. Liu and D. Wang, NIPS 2016], a deterministic
algorithm for sampling from a given probability density with unknown
normalization. We prove that in the large particle limit the empirical measure
of the particle system converges to a solution of a non-local and nonlinear
PDE. We also prove global existence, uniqueness and regularity of the solution
to the limiting PDE. Finally, we prove that the solution to the PDE converges
to the unique invariant solution in long time limit.

该研究通过研究与 Stein 变分梯度下降相关的相互作用粒子系统，在大粒子极限下，粒子系统的经验测量收敛于非局部和非线性 PDE 的解，并证明此限制 PDE 的解的全局存在、唯一性和正则性，最终证明了 PDE 的解在长期限制下收敛于唯一的不变解。