A Family of MCMC Methods on Implicitly Defined Manifolds

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v22-brubaker12, title = {A Family of MCMC Methods on Implicitly Defined Manifolds}, author = {Brubaker, Marcus and Salzmann, Mathieu and Urtasun, Raquel}, booktitle = {Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics}, pages = {161--172}, year = {2012}, editor = {Lawrence, Neil D. and Girolami, Mark}, volume = {22}, series = {Proceedings of Machine Learning Research}, address = {La Palma, Canary Islands}, month = {21--23 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v22/brubaker12/brubaker12.pdf}, url = {https://proceedings.mlr.press/v22/brubaker12.html}, 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.} }
Endnote
%0 Conference Paper %T A Family of MCMC Methods on Implicitly Defined Manifolds %A Marcus Brubaker %A Mathieu Salzmann %A Raquel Urtasun %B Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2012 %E Neil D. Lawrence %E Mark Girolami %F pmlr-v22-brubaker12 %I PMLR %P 161--172 %U https://proceedings.mlr.press/v22/brubaker12.html %V 22 %X 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.
RIS
TY - CPAPER TI - A Family of MCMC Methods on Implicitly Defined Manifolds AU - Marcus Brubaker AU - Mathieu Salzmann AU - Raquel Urtasun BT - Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics DA - 2012/03/21 ED - Neil D. Lawrence ED - Mark Girolami ID - pmlr-v22-brubaker12 PB - PMLR DP - Proceedings of Machine Learning Research VL - 22 SP - 161 EP - 172 L1 - http://proceedings.mlr.press/v22/brubaker12/brubaker12.pdf UR - https://proceedings.mlr.press/v22/brubaker12.html AB - 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. ER -
APA
Brubaker, M., Salzmann, M. & Urtasun, R.. (2012). A Family of MCMC Methods on Implicitly Defined Manifolds. Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 22:161-172 Available from https://proceedings.mlr.press/v22/brubaker12.html.

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