# Evaluating the Implicit Midpoint Integrator for Riemannian Manifold Hamiltonian Monte Carlo

@inproceedings{Brofos2021EvaluatingTI, title={Evaluating the Implicit Midpoint Integrator for Riemannian Manifold Hamiltonian Monte Carlo}, author={James A. Brofos and Roy R. Lederman}, year={2021} }

Riemannian manifold Hamiltonian Monte Carlo is traditionally carried out using the generalized leapfrog integrator. However, this integrator is not the only choice and other integrators yielding valid Markov chain transition operators may be considered. In this work, we examine the implicit midpoint integrator as an alternative to the generalized leapfrog integrator. We discuss advantages and disadvantages of the implicit midpoint integrator for Hamiltonian Monte Carlo, its theoretical…

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## References

SHOWING 1-10 OF 29 REFERENCES

Riemann manifold Langevin and Hamiltonian Monte Carlo methods

- Computer Science
- 2011

The methodology proposed automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density, and substantial improvements in the time‐normalized effective sample size are reported when compared with alternative sampling approaches.

Implicit Hamiltonian Monte Carlo for Sampling Multiscale Distributions

- Computer Science
- 2019

This work provides intuition as well as a formal analysis showing how these multiscale distributions limit the stepsize of leapfrog and shows how the implicit midpoint method can be used, together with Newton-Krylov iteration, to circumvent this limitation and achieve major efficiency gains.

A General Metric for Riemannian Manifold Hamiltonian Monte Carlo

- MathematicsGSI
- 2013

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

MCMC Using Hamiltonian Dynamics

- Physics
- 2011

Hamiltonian dynamics can be used to produce distant proposals for the Metropolis algorithm, thereby avoiding the slow exploration of the state space that results from the diffusive behaviour of…

Geodesic Monte Carlo on Embedded Manifolds

- MathematicsScandinavian 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.

Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations

- Physics
- 2004

New Insights and Perspectives on the Natural Gradient Method

- Computer ScienceJ. Mach. Learn. Res.
- 2020

This paper critically analyze this method and its properties, and shows how it can be viewed as a type of approximate 2nd-order optimization method, where the Fisher information matrix can be view as an approximation of the Hessian.

A Family of MCMC Methods on Implicitly Defined Manifolds

- Mathematics, Computer ScienceAISTATS
- 2012

A general constrained version of Hamiltonian Monte Carlo is proposed, and conditions under which the Markov chain is convergent are given, which define a family of MCMC methods which can be applied to sample from distributions on non-linear manifolds.

Information geometry and its applications

- Mathematics2012 9th European Radar Conference
- 2012

Information geometry is a new mathematical discipline which applies the methodology of differential geometry to statistics. Therefore, families of exponential distributions are considered as embedded…

Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems

- Physics
- 1999

This second edition has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises.