• Corpus ID: 231925157

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… 
Accelerating Hamiltonian Monte Carlo via Chebyshev Integration Time
TLDR
This work proposes a scheme of time-varying integration time based on the roots of Chebyshev polynomials for Hamiltonian Monte Carlo (HMC) and shows that in the case of quadratic potential f, ideal HMC with this choice of integration time only takes O ( √ κ log 1 (cid:15) ) number of iterations to reach Wasserstein-2 distance less than (cids:15).
Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions
TLDR
A delayed rejection variant of Hamiltonian Monte Carlo that makes one or more subsequent proposals each using a step size geometrically smaller than the last if an initial HMC trajectory is rejected, providing increased robustness to step size misspecification.

References

SHOWING 1-10 OF 29 REFERENCES
Riemann manifold Langevin and Hamiltonian Monte Carlo methods
TLDR
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
TLDR
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
TLDR
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
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
TLDR
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.
New Insights and Perspectives on the Natural Gradient Method
TLDR
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
TLDR
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
  • F. Opitz
  • Mathematics
    2012 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
TLDR
This second edition has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises.
...
...