• Corpus ID: 88518298

# Hamiltonian Monte Carlo on Symmetric and Homogeneous Spaces via Symplectic Reduction

@article{Barp2019HamiltonianMC,
title={Hamiltonian Monte Carlo on Symmetric and Homogeneous Spaces via Symplectic Reduction},
author={Alessandro Barp and Anthony D. Kennedy and Mark A. Girolami},
journal={arXiv: Computation},
year={2019}
}
• Published 7 March 2019
• Mathematics
• arXiv: Computation
The Hamiltonian Monte Carlo method generates samples by introducing a mechanical system that explores the target density. For distributions on manifolds it is not always simple to perform the mechanics as a result of the lack of global coordinates, the constraints of the manifold, and the requirement to compute the geodesic flow. In this paper we explain how to construct the Hamiltonian system on naturally reductive homogeneous spaces using symplectic reduction, which lifts the HMC scheme to a…
8 Citations

## Figures from this paper

### Hamiltonian Monte Carlo on Lie Groups and Constrained Mechanics on Homogeneous Manifolds

The correspondence between the various formulations of Hamiltonian mechanics on Lie groups, and their induced HMC algorithms is obtained, and how mechanics on homogeneous spaces can be formulated as a constrained system over their associated Lie groups is explained.

### Optimization on manifolds: A symplectic approach

• Mathematics
• 2021
There has been great interest in using tools from dynamical systems and numerical analysis of differential equations to understand and construct new optimization methods. In particular, recently a

### A Riemann–Stein kernel method

• Mathematics
Bernoulli
• 2022
This paper proposes and studies a numerical method for approximation of posterior expectations based on interpolation with a Stein reproducing kernel. Finite-sample-size bounds on the approximation

### Irreversible Langevin MCMC on Lie Groups

• Mathematics
GSI
• 2019
An irreversible HMC-like MCMC algorithm on $\mathcal G$, where the momentum is updated by solving an OU process on the corresponding Lie algebra $\mathfrak g$ and the Hamiltonian system is approximate with a reversible symplectic integrator followed by a Metropolis-Hastings correction step.

### PR ] 6 M ay 2 02 1 A UNIFYING AND CANONICAL DESCRIPTION OF MEASURE-PRESERVING

• Mathematics
• 2021
A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. In this paper, we develop a geometric theory that

### Diffusion bridges for stochastic Hamiltonian systems with applications to shape analysis

• Mathematics
ArXiv
• 2020
This work develops a general scheme for bridge sampling in the case of finite dimensional systems of shape landmarks and singular solutions in fluid dynamics, that covers stochastic landmark models for which no suitable simulation method has been proposed in the literature.

### Diffusion Bridges for Stochastic Hamiltonian Systems and Shape Evolutions

• Physics
SIAM J. Imaging Sci.
• 2022
This work demonstrates how to apply state-of-the-art diffusion bridge simulation methods to recently introduced stochastic shape deformation models thereby substantially expanding the applicability of such models.

## References

SHOWING 1-10 OF 55 REFERENCES

### Geodesic Monte Carlo on Embedded Manifolds

• Mathematics
Scandinavian journal of statistics, theory and applications
• 2013
Methods to simulate from probability distributions that themselves are defined on a manifold, with common examples being classes of distributions describing directional statistics, are considered.

### Generalizing the No-U-Turn Sampler to Riemannian Manifolds

Hamiltonian Monte Carlo provides efficient Markov transitions at the expense of introducing two free parameters: a step size and total integration time. Because the step size controls discretization

### Geodesic Lagrangian Monte Carlo over the space of positive definite matrices: with application to Bayesian spectral density estimation

• Mathematics
Journal of statistical computation and simulation
• 2018
ABSTRACT We present geodesic Lagrangian Monte Carlo, an extension of Hamiltonian Monte Carlo for sampling from posterior distributions defined on general Riemannian manifolds. We apply this new

### Shadow Hamiltonians, Poisson brackets, and gauge theories

• Physics
• 2013
Numerical lattice gauge theory computations to generate gauge field configurations including the effects of dynamical fermions are usually carried out using algorithms that require the molecular

### A General Metric for Riemannian Manifold Hamiltonian Monte Carlo

A new metric for RMHMC is proposed without limitations and its success on a distribution that emulates many hierarchical and latent models is verified.

### POISSON STRUCTURES ON THE COTANGENT BUNDLE OF A LIE GROUP OR A PRINCIPLE BUNDLE AND THEIR REDUCTIONS

• Mathematics
• 1993
On a cotangent bundle T*G of a Lie group G one can describe the standard Liouville form θ and the symplectic form dθ in terms of the right Maurer Cartan form and the left moment mapping (of the right

### A symplectic integrator for riemannian manifolds

• Mathematics
• 1996
SummaryThe configuration spaces of mechanical systems usually support Riemannian metrics which have explicitly solvable geodesic flows and parallel transport operators. While not of primary interest,

### An Ergodic Sampling Scheme for Constrained Hamiltonian Systems with Applications to Molecular Dynamics

Abstract This article addresses the problem of computing the Gibbs distribution of a Hamiltonian system that is subject to holonomic constraints. In doing so, we extend recent ideas of Cancès et al.

### Differential Geometry, Lie Groups, and Symmetric Spaces

Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric