Positive Curvature and Hamiltonian Monte Carlo
نویسندگان
چکیده
The Jacobi metric introduced in mathematical physics can be used to analyze Hamiltonian Monte Carlo (HMC). In a geometrical setting, each step of HMC corresponds to a geodesic on a Riemannian manifold with a Jacobi metric. Our calculation of the sectional curvature of this HMC manifold allows us to see that it is positive in cases such as sampling from a high dimensional multivariate Gaussian. We show that positive curvature can be used to prove theoretical concentration results for HMC Markov chains.
منابع مشابه
at / 9 80 70 30 v 1 1 3 Ju l 1 99 8 Simulation of an Abelian Higgs theory in world line ( polymer ) representation
Simulation of an Abelian Higgs theory in world line (polymer) representation. In World Line Path Integral representation Abelian Higgs theories admit a curvature term in the Hamiltonian. Using Monte Carlo simulations we show that the curvature term drives a phase transition at sufficiently strong coupling. In quenched approximation this phase transition is smooth, with implications that the mod...
متن کاملSurface tension in an intrinsic curvature model with fixed one-dimensional boundaries
A triangulated fixed connectivity surface model is investigated by using the Monte Carlo simulation technique. In order to have the macroscopic surface tension τ , the vertices on the one-dimensional boundaries are fixed as the edges (=circles) of the tubular surface in the simulations. The size of the tubular surface is chosen such that the projected area becomes the regular square of area A. ...
متن کاملEnsemble projector Monte Carlo method, studying the lattice Schwinger model in the Hamiltonian formulation.
The ensemble projector Monte Carlo method is a promising method to study lattice gauge theories with fermions in the Hamiltonian formulation. We study the massive Schwinger model and show, that consistent results are obtained in the presence of positive and negative matrix elements. The expectation values for the average energy calculated from matrix elements with negative and positive scores, ...
متن کاملCurvature, Concentration and Error Estimates for Markov Chain Monte Carlo by Aldéric Joulin
We provide explicit nonasymptotic estimates for the rate of convergence of empirical means of Markov chains, together with a Gaussian or exponential control on the deviations of empirical means. These estimates hold under a “positive curvature” assumption expressing a kind of metric ergodicity, which generalizes the Ricci curvature from differential geometry and, on finite graphs, amounts to co...
متن کاملMATHEMATICAL ENGINEERING TECHNICAL REPORTS Hamiltonian Monte Carlo with Explicit, Reversible, and Volume-preserving Adaptive Step Size Control
Hamiltonian Monte Carlo is a Markov chain Monte Carlo method that uses Hamiltonian dynamics to efficiently produce distant samples. It employs geometric numerical integration to simulate Hamiltonian dynamics, which is a key of its high performance. We present a Hamiltonian Monte Carlo method with adaptive step size control to further enhance the efficiency. We propose a new explicit, reversible...
متن کامل