用微分方程近似测地线，提升复杂多峰分布采样效率
Geodesic Slice Sampler for Multimodal Distributions with Strong Curvature
- 通过求解微分方程近似测地线，拓展切片采样适用范围
- 在强曲率与模式间快速切换的分布中表现更优
- 适合高维复杂分布采样，如生物物理或贝叶斯推断
传统马尔可夫链蒙特卡洛采样方法在尖锐曲率、复杂几何结构及多峰分布上表现不佳。切片采样可缓解局部探索低效问题，黎曼几何有助于处理尖锐曲率。近期扩展实现了在黎曼流形上的切片采样，但受限于测地线需有闭式解。本文提出一种将命中-运行切片采样推广至更一般几何结构的方法，通过将测地线近似为微分方程的解，使采样能有效探索强曲率区域，并在多峰分布中实现模式间的快速跃迁。我们在具有挑战性的采样任务中验证了该方法的优势。
原文摘要 · Abstract (English)
Traditional Markov Chain Monte Carlo sampling methods often struggle with sharp curvatures, intricate geometries, and multimodal distributions. Slice sampling can resolve local exploration inefficiency issues, and Riemannian geometries help with sharp curvatures. Recent extensions enable slice sampling on Riemannian manifolds, but they are restricted to cases where geodesics are available in a closed form. We propose a method that generalizes Hit-and-Run slice sampling to more general geometries tailored to the target distribution, by approximating geodesics as solutions to differential equations. Our approach enables the exploration of the regions with strong curvature and rapid transitions between modes in multimodal distributions. We demonstrate the advantages of the approach over challenging sampling problems.
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