Geodesic slice sampling on the sphere

TL;DR

This work proposes a shrinkage based and an idealized geodesic slice sampling Markov chain, designed to generate approximate samples from distributions on the sphere, and proves that under weak regularity conditions geodesic slice sampling is uniformly ergodic.

Abstract

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.

Figures (10)

• Figure 1: Transition mechanism of the ideal geodesic slice sampler. (a) Sample a random level $t$ at the current point $x$. (b) Sample a random point $v$ on ${\mathbb{S}^{d -2}_{x}}.$ (c) Sample uniformly from the geodesic level set $L(x,v,t)$.
• Figure 2: Shrinkage procedure of the geodesic shrinkage slice sampler. (a) First proposal is rejected. (b) Second proposal is rejected and becomes new right bound of the proposal interval. (c) Third proposal is rejected and becomes new left bound of the proposal interval. (d) Fourth proposal lies in geodesic level set $L(x,v,t)$ and is accepted.
• Figure 3: Rigid registration of point clouds. Each point cloud consists of 214 carbon-$\alpha$ atom positions. Points are colored according to their sequence position: N-terminal amino acids are shown in blue, C-terminal points are red. (A) Target point cloud showing AK in closed configuration. (B) Source point cloud showing AK in open configuration and in an orientation that differs from the orientation of the target. Note that target and source points with identical color correspond to each other. However, due to the non-rigid conformational change not all source points will generate target points. (C) Source rotated optimally so as to generate the target with high probability according to the CPD model (\ref{['eq:cpd']}). Note that the points in the source that do not generate a target point are shown in gray. These are located in the moveable domains of the AK structure (for example the orange domain that is closed in the target, but open in the source).
• Figure 4: Probabilistic rigid registration of the open and closed structure of adenylate kinase based on posterior $p$ given in \ref{['eq:posterior-cpd']}. (A) Slice through $p$ along a great circle $x(\theta) = \cos\theta\, x_1 + \sin\theta\, x_2$ where $x_1\in\mathbb S^{3}$ is the global optimum and $x_2\in{\mathbb{S}^{2}_{x_1}}$. (B) Marginal distribution of log posterior probabilities $\log p(x_i)$ based on a discretization of $\mathbb S^{3}$ using $\sim 1.28 \times 10^6$ regularly placed unit quaternions $x_i \in \mathbb S^{3}$. (C) Success rate against number of Markov chain iterations for all four samplers. (D) Success rate against computation time.
• Figure 5: Marginal distributions (shown in colors) for the mixture of vMF distributions ($d=10, \kappa=100, K=5$) where columns corresponds to MCMC methods and rows correspond to the dimension of the marginal (only the first two dimensions are shown). Here, $\{e_i\}_{i=1}^{10}$ denotes the standard basis in $\mathbb R^{10}$. The black dashed line indicates the true marginals computed analytically for the mixture of vMF distribution.

Theorems & Definitions (24)

• Lemma 1
• Lemma 2
• Lemma 3
• Remark 4
• Proposition 5
• Theorem 6
• Remark 7
• Remark 8
• Remark 9
• Lemma 10: MeynTweedie