A Family of MCMC Methods on Implicitly Defined Manifolds
Marcus Brubaker, Mathieu Salzmann, Raquel Urtasun ; JMLR W&CP 22: 161-172, 2012.
Abstract
Traditional MCMC methods are only applicable to distributions which can be defined on $\mathbb{R}^n$. However, there exist many application domains where the distributions cannot easily be defined on a Euclidean space. To address this limitation, we propose a general constrained version of Hamiltonian Monte Carlo, and give conditions under which the Markov chain is convergent. Based on this general framework we define a family of MCMC methods which can be applied to sample from distributions on non-linear manifolds. We demonstrate the effectiveness of our approach on a variety of problems including sampling from the Bingham-von Mises-Fisher distribution, collaborative filtering and human pose estimation.
