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.

翻译：摘要：根据未归一化的密度对概率分布进行积分是贝叶斯推断及其他领域的核心任务。我们引入新方法，通过极小化与目标分布的最大平均差异（MMD）的交互粒子系统，构建由少量加权样本组成的近似期望的求积规则。这些方法将经典均值漂移算法以及近期用于经验分布最优量化的算法扩展到连续分布情形。关键在于，我们的方法构建了对于未知归一化常数不变的MMD极小化动力学，且同时支持无梯度与含梯度的实现。由此得到的均值漂移交互粒子系统收敛迅速，能够捕捉各向异性与多模态性，避免模态坍塌，并可扩展至高维。我们在广泛基准采样问题上验证了其性能，包括多模态混合分布、贝叶斯分层模型、偏微分方程约束反问题等。