On the unconventional Hug integrator

TL;DR

This paper generalizes the Hug integrator from hypersurfaces to arbitrary manifolds, analyzing its convergence and uncovering unique dynamical properties, including a supraconvergence phenomenon.

Contribution

It introduces a novel interpretation of Hug as a discretization of complex differential systems and proves its convergence with a surprising second-order accuracy.

Findings

Hug can be extended to arbitrary manifolds.

A supraconvergence property is established for the discretization.

Some Hug trajectories do not cover the entire manifold.

Abstract

Hug is a recently proposed iterative mapping used to design efficient updates in Markov chain Monte Carlo (MCMC) methods. Hug generates proposals that remain very close to hypersurfaces (level sets) of constant probabilty density. We analyse a generalization of Hug from hypersurfaces to manifolds of arbitrary dimensions, not necessarily arising in a sampling context. The analysis is based on interpreting, in a nonstandard way, Hug as a consistent discretization of a system of differential equations with a rather complicated structure. The proof of convergence of this discretization includes a number of unusual features we explore fully, in particular a supraconvergence property is established, whereby second order of convergence is attained with consistency of the first order. We uncover and discuss an unexpected property of the solutions of the underlying dynamical system that manifest itself by the existence of Hug trajectories that fail to cover the manifold of interest.