用低秩混合模型提升高维重要性采样效率
Scalable Importance Sampling in High Dimensions with Low-Rank Mixture Proposals
- 采用低秩混合主成分分析模型作为采样提案分布
- 在三个模拟系统中实现更高效、更准确的罕见事件估计
- 适合需要高效处理高维稀有事件的工程与科研场景
重要性采样是一种蒙特卡洛技术，通过偏向目标稀有事件的采样分布，高效估计罕见事件的概率。通过从学习得到的提案分布中抽取加权样本，重要性采样可实现对稀有事件或分布尾部的更高效估计。常用提案密度为高斯混合模型（GMM），但在高维空间中估计全秩协方差矩阵存在数值不稳定性问题。本文提出使用概率主成分分析混合模型（MPPCA）作为重要性采样方法的参数化提案分布。MPPCA 是一种低秩混合模型，可通过期望最大化算法快速拟合，即使在高维空间中亦然。我们在三个模拟系统上验证该方法，结果表明其在样本效率和失效分布刻画质量上均有持续提升。
原文摘要 · Abstract (English)
Importance sampling is a Monte Carlo technique for efficiently estimating the likelihood of rare events by biasing the sampling distribution towards the rare event of interest. By drawing weighted samples from a learned proposal distribution, importance sampling allows for more sample-efficient estimation of rare events or tails of distributions. A common choice of proposal density is a Gaussian mixture model (GMM). However, estimating full-rank GMM covariance matrices in high dimensions is a challenging task due to numerical instabilities. In this work, we propose using mixtures of probabilistic principal component analyzers (MPPCA) as the parametric proposal density for importance sampling methods. MPPCA models are a type of low-rank mixture model that can be fit quickly using expectation-maximization, even in high-dimensional spaces. We validate our method on three simulated systems, demonstrating consistent gains in sample efficiency and quality of failure distribution characterization.
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