Hierarchical Riemannian manifold Hamiltonian Monte Carlo algorithms

2026-04-10

AI Summary

This work addresses the computational expense and implementation challenges of Riemannian Manifold Hamiltonian Monte Carlo (RMHMC) for sampling from high-dimensional, complex probability distributions. The authors propose an adaptive hierarchical RMHMC method that introduces a hierarchical structure into the mass matrix to effectively capture the local geometry of the target distribution. They develop, for the first time, a closed-form explicit Leapfrog integrator tailored to hierarchical RMHMC, enabling seamless integration with dynamic HMC frameworks such as NUTS. Furthermore, they design an automatic tuning mechanism that adaptively optimizes the mass matrix without requiring the target density itself to exhibit hierarchical structure. Empirical results demonstrate that the proposed method significantly improves sampling efficiency and practical performance in high-dimensional Bayesian inference tasks.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Hamiltonian Monte Carlo (HMC) and its dynamic extensions, such as the No-U-Turn Sampler (NUTS), are powerful Markov chain Monte Carlo methods for sampling from complex, high-dimensional probability distributions. Riemannian manifold Hamiltonian Monte Carlo (RMHMC) extends HMC by allowing the mass matrix to depend on position, which can substantially improve mixing but also makes implementation considerably more challenging. In this paper, we study an adaptive hierarchical version of RMHMC that is well suited to many hierarchical sampling problems. A key feature of hierarchical RMHMC is that, unlike general RMHMC, it admits a closed-form explicit leapfrog integrator, enabling efficient implementation and direct use within dynamic HMC methods such as NUTS. We introduce an adaptive scheme that automatically tunes the parameters of the hierarchical mass matrix during simulation. Importantly, the target density need not exhibit any hierarchical or block structure; the hierarchy is instead imposed on the mass matrix as a modeling device to capture the local geometry of the target distribution. Numerical experiments demonstrate appealing empirical performance in high-dimensional Bayesian inference problems.

Problem

• Hamiltonian Monte Carlo
• Riemannian manifold
• hierarchical sampling
• high-dimensional Bayesian inference
• mass matrix

Innovation

• Hierarchical RMHMC
• Explicit leapfrog integrator
• Adaptive mass matrix
• Riemannian manifold HMC
• Dynamic HMC