用高斯混合逼近法高效采样无归一化目标分布。
Sampling via Gaussian Mixture Approximations
- 两阶段采样：先初始化高斯分量，再通过样本优化权重或参数。
- 仅需无归一化密度评估，无需梯度，计算高效且结果准确。
- 适合需要快速、无梯度采样的科研与工程场景。
我们提出一类用于无归一化目标分布采样的高斯混合逼近（GMA）采样器，包括仅优化权重的W-GMA、拉普拉斯混合逼近（LMA）、期望最大化GMA（EM-GMA）等变体。GMA采用简单两阶段范式：(i) 初始化有限个高斯分量并从提议混合分布中采样；(ii) 通过基于样本的KL散度目标函数优化成分权重，或同时优化均值与方差，该目标仅需无归一化密度评估，随后进行分层重采样。方法完全无梯度，计算高效：利用高斯采样便捷性、高效的优化方法（投影梯度下降、镜像下降、EM）以及分层重采样对目标分布的鲁棒性，生成忠实于目标的样本。我们在温和条件下证明该优化-重采样方案可产生一致逼近，并通过实证结果验证其在多种分布上的精度与速度优势。
原文摘要 · Abstract (English)
We present a family of \textit{Gaussian Mixture Approximation} (GMA) samplers for sampling unnormalised target densities, encompassing \textit{weights-only GMA} (W-GMA), \textit{Laplace Mixture Approximation} (LMA), \textit{expectation-maximization GMA} (EM-GMA), and further variants. GMA adopts a simple two-stage paradigm: (i) initialise a finite set of Gaussian components and draw samples from a proposal mixture; (ii) fit the mixture to the target by optimising either only the component weights or also the means and variances, via a sample-based KL divergence objective that requires only evaluations of the unnormalised density, followed by stratified resampling. The method is gradient-free, and computationally efficient: it leverages the ease of sampling from Gaussians, efficient optimisation methods (projected gradient descent, mirror descent, and EM), and the robustness of stratified resampling to produce samples faithful to the target. We show that this optimisation-resampling scheme yields consistent approximations under mild conditions, and we validate this methodology with empirical results demonstrating accuracy and speed across diverse densities.
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