球面上的测地切片采样
Geodesic slice sampling on the sphere
AI总结：
本文提出了一种高效的球面上测地切片采样方法，通过收缩机制实现高效采样，且在复杂分布上优于传统采样器。
摘要：
球面上的概率测度是一个重要的统计模型类别，例如用于建模方向数据或形状。由于其广泛应用，以及作为算法的基本构建块，对球面上分布的有效采样具有高度的期望。我们提出了一种基于收缩和理想化的测地切片采样马尔可夫链，旨在生成球面上分布的近似样本。特别是，该算法的收缩版本可以实现，使其运行高效且无须调参。我们验证了可逆性，并证明在弱正则条件下，测地切片采样是均匀遍历的。数值实验显示，所提出的切片采样器在具有挑战性的目标上表现出色，包括出现在刚体配准问题中的分布以及von Mises-Fisher分布的混合物。在这些设置中，我们的方法优于标准采样器，如随机游走Metropolis-Hastings和Hamiltonian Monte Carlo。
英文摘要：
Probability measures on the sphere form an important class of statistical models and are used, for example, in modeling directional data or shapes. Due to their widespread use, but also as an algorithmic building block, efficient sampling of distributions on the sphere is highly desirable. We propose a shrinkage based and an idealized geodesic slice sampling Markov chain, designed to generate approximate samples from distributions on the sphere. In particular, the shrinkage-based version of the algorithm can be implemented such that it runs efficiently and has no tuning parameters. We verify reversibility and prove that under weak regularity conditions geodesic slice sampling is uniformly ergodic. Numerical experiments show that the proposed slice samplers achieve excellent mixing on challenging targets including distributions arising in rigid-registration problems and mixtures of von Mises-Fisher distributions. In these settings our approach outperforms standard samplers such as random-walk Metropolis-Hastings and Hamiltonian Monte Carlo.
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