A Family of MCMC Methods on Implicitly Defined Manifolds

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Marcus Brubaker, Mathieu Salzmann, Raquel Urtasun ;
Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, PMLR 22:161-172, 2012.

Abstract

Traditional MCMC methods are only applicable to distributions which can be defined on \mathbbR^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.

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