1,151 research outputs found
Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation
We give the first rigorous proof of the convergence of Riemannian Hamiltonian
Monte Carlo, a general (and practical) method for sampling Gibbs distributions.
Our analysis shows that the rate of convergence is bounded in terms of natural
smoothness parameters of an associated Riemannian manifold. We then apply the
method with the manifold defined by the log barrier function to the problems of
(1) uniformly sampling a polytope and (2) computing its volume, the latter by
extending Gaussian cooling to the manifold setting. In both cases, the total
number of steps needed is O^{*}(mn^{\frac{2}{3}}), improving the state of the
art. A key ingredient of our analysis is a proof of an analog of the KLS
conjecture for Gibbs distributions over manifolds
Information-geometric Markov Chain Monte Carlo methods using Diffusions
Recent work incorporating geometric ideas in Markov chain Monte Carlo is
reviewed in order to highlight these advances and their possible application in
a range of domains beyond Statistics. A full exposition of Markov chains and
their use in Monte Carlo simulation for Statistical inference and molecular
dynamics is provided, with particular emphasis on methods based on Langevin
diffusions. After this geometric concepts in Markov chain Monte Carlo are
introduced. A full derivation of the Langevin diffusion on a Riemannian
manifold is given, together with a discussion of appropriate Riemannian metric
choice for different problems. A survey of applications is provided, and some
open questions are discussed.Comment: 22 pages, 2 figure
Curvature and Concentration of Hamiltonian Monte Carlo in High Dimensions
In this article, we analyze Hamiltonian Monte Carlo (HMC) by placing it in
the setting of Riemannian geometry using the Jacobi metric, so that each step
corresponds to a geodesic on a suitable Riemannian manifold. We then combine
the notion of curvature of a Markov chain due to Joulin and Ollivier with the
classical sectional curvature from Riemannian geometry to derive error bounds
for HMC in important cases, where we have positive curvature. These cases
include several classical distributions such as multivariate Gaussians, and
also distributions arising in the study of Bayesian image registration. The
theoretical development suggests the sectional curvature as a new diagnostic
tool for convergence for certain Markov chains.Comment: Comments welcom
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