Dimension-independent Markov chain Monte Carlo on the sphere

TL;DR

This work develops dimension-independent Markov chain Monte Carlo methods for sampling posteriors on high-dimensional spheres with angular central Gaussian priors. By lifting the problem to an ambient Hilbert space and using push-forward reprojection kernels, the authors construct two reversible sphere samplers—the reprojected pCN-MH and the reprojected ESS—that inherit favorable convergence properties from their ambient-space counterparts. They prove uniform ergodicity in finite dimensions and provide theoretical arguments for dimension-independent spectral gaps, complemented by numerical experiments on Bayesian binary level-set inversion and Bayesian density estimation that demonstrate robust performance as dimension grows. The methods enable efficient Bayesian inference on directional data on spheres and offer a pathway to extend to other priors, with potential impact on inverse problems and directional statistics in infinite-dimensional settings.

Abstract

We consider Bayesian analysis on high-dimensional spheres with angular central Gaussian priors. These priors model antipodally symmetric directional data, are easily defined in Hilbert spaces and occur, for instance, in Bayesian binary classification and level set inversion. In this paper we derive efficient Markov chain Monte Carlo methods for approximate sampling of posteriors with respect to these priors. Our approaches rely on lifting the sampling problem to the ambient Hilbert space and exploit existing dimension-independent samplers in linear spaces. By a push-forward Markov kernel construction we then obtain Markov chains on the sphere, which inherit reversibility and spectral gap properties from samplers in linear spaces. Moreover, our proposed algorithms show dimension-independent efficiency in numerical experiments.

Figures (8)

• Figure 3.1: Illustration of the steps for drawing states using the naıve reprojection kernel $\bar{K}$ in \ref{['eq:naive']} for $\mathbb{X}=\mathbb{H}$, $\mathbb{Y}=\mathbb{S}$, and $T=\Pi^{\mathbb{S}}$ in \ref{['eq:radial_projection_map_T']}. Starting from $\bar{x}$, an intermediate state $y\in \mathbb{H}$ is drawn from the $\nu$-reversible transition kernel $K(\bar{x},\hbox{$\cdot$})$ on $\mathbb{H}$. The next state drawn from the naïve reprojection kernel $\bar{K}(\bar{x},\hbox{$\cdot$})$ is then $\bar{y} \coloneqq T(y)=\Pi^{\mathbb{S}}(y)$. Solid arrows indicate deterministic maps, whereas dashed arrows indicate randomised maps, i.e. draws from transition kernels.
• Figure 3.2: Comparison of marginals of $\mu = \Pi^{\mathbb{S}}_\sharp\nu$ (target), $\mu\bar{K}$ (naïve reprojection) with $\bar{K}$ as in \ref{['eq:naive']}, and $\mu(\Pi^{\mathbb{S}}_{\sharp} K)$ (reprojection) with $\Pi^{\mathbb{S}}_{\sharp} K$ as in \ref{['eq:TM_gen']} with pCN-kernel $K$ and $T=\Pi^{\mathbb{S}}$.
• Figure 3.3: Illustration of the steps in the reprojection method from \ref{['sec:reprojection_method']}. The reprojection method defines a $\mu$-reversible transition kernel on $\mathbb{S}$ in terms of a $\nu$-reversible transition kernel $K$ on the ambient space $\mathbb{H}$, where $\mathbb{S} \coloneqq T(\mathbb{H})$, $\mu \coloneqq T_{\sharp} \nu$, $\nu_{|T} (\bar{x}, \hbox{$\cdot$})$ is the regular conditional distribution of $X \sim \nu$ given $T(X) = \bar{x}$, and $T=\Pi^{\mathbb{S}}$. Solid arrows indicate deterministic maps, whereas dashed arrows indicate randomised maps, i.e. draws from transition kernels.
• Figure 4.1: True $g^\dagger$, $u^\dagger$, $p^\dagger$, and true observations $o^\dagger = (p^\dagger(0.2),\ldots,p^\dagger(0.8))^\top$.
• Figure 4.2: Prior, likelihood and posterior for dimension $d=3$; the red spot indicates the (projected) truth $\bar{x}^\dagger$.