用粒子系统高效逼近复杂分布的期望值，避免归一化难题。
To discretize continually: Mean shift interacting particle systems for Bayesian inference
- 基于均值漂移的粒子系统最小化分布差异
- 高维多模态分布下快速收敛，不发生模式崩溃
- 无需梯度或可结合梯度，适合复杂贝叶斯推断
在贝叶斯推断等场景中，对给定非归一化密度的概率分布进行积分是核心任务。本文提出新方法，通过一个交互粒子系统构造少量加权样本的求积规则，以最小化最大均值差异（MMD）逼近目标分布。该方法扩展经典均值漂移算法及最近的最优量化算法，适用于连续分布。关键优势在于：动态过程对未知归一化常数不变，支持无梯度与有梯度两种实现方式。所提均值漂移交互粒子系统收敛迅速，能捕捉各向异性与多模态特性，避免模式崩溃，且可扩展至高维。在多种基准测试中表现优异，涵盖多模混合、贝叶斯层次模型、偏微分方程约束反问题等。
原文摘要 · Abstract (English)
Integration against a probability distribution given its unnormalized density is a central task in Bayesian inference and other fields. We introduce new methods for approximating such expectations with a small set of weighted samples -- i.e., a quadrature rule -- constructed via an interacting particle system that minimizes maximum mean discrepancy (MMD) to the target distribution. These methods extend the classical mean shift algorithm, as well as recent algorithms for optimal quantization of empirical distributions, to the case of continuous distributions. Crucially, our approach creates dynamics for MMD minimization that are invariant to the unknown normalizing constant; they also admit both gradient-free and gradient-informed implementations. The resulting mean shift interacting particle systems converge quickly, capture anisotropy and multi-modality, avoid mode collapse, and scale to high dimensions. We demonstrate their performance on a wide range of benchmark sampling problems, including multi-modal mixtures, Bayesian hierarchical models, PDE-constrained inverse problems, and beyond.
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