Abstract
It is well-known that irreversible MCMC algorithms converge faster to their stationary distributions than reversible ones. Using the special geometric structure of Lie groups \(\mathcal G\) and dissipation fields compatible with the symplectic structure, we construct an irreversible HMC-like MCMC algorithm on \(\mathcal G\), where we first update the momentum by solving an OU process on the corresponding Lie algebra \(\mathfrak g\), and then approximate the Hamiltonian system on \(\mathcal G\times \mathfrak g\) with a reversible symplectic integrator followed by a Metropolis-Hastings correction step. In particular, when the OU process is simulated over sufficiently long times, we recover HMC as a special case. We illustrate this algorithm numerically using the example \(\mathcal G= SO(3)\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For a non-matrix group, simply replace \(\mathrm {Tr}\left( \partial _x V^T \bar{g}_k \xi _i\right) \) with \(e_i|_g(V)\).
References
Arnaudon, Alexis, De Castro, A.L., Holm, D.D.: Noise and dissipation on coadjoint orbits. J. Nonlinear Sci. 28(1), 91–145 (2018)
Arnaudon, A., Takao, S., Networks of coadjoint orbits: from geometric to statistical mechanics, arXiv preprint arXiv:1804.11139 (2018)
Barp, A.: Hamiltonian Monte Carlo On Lie Groups and Constrained Mechanics on Homogeneous Manifolds, arXiv preprint arXiv:1903.04662 (2019)
Barp, Alessandro, Briol, François-Xavier, Kennedy, A.D., Girolami, M.: Geometry and dynamics for markov chain monte carlo. Annu. Rev. Stat. Appl. 5, 451–471 (2018)
Barp, A., Kennedy, A.D., Girolami, M.: Hamiltonian Monte Carlo on Symmetric and Homogeneous Spaces via Symplectic Reduction, arXiv preprint arXiv:1903.02699 (2019)
Betancourt, Michael, Byrne, Simon, Livingstone, Sam, Girolami, Mark, et al.: The geometric foundations of hamiltonian monte carlo. Bernoulli 23(4A), 2257–2298 (2017)
Bismut, J.M.: Méecanique aléeatoire lect. Notes Maths 966, 19881 (1981)
Bou-Rabee, N., Vanden-Eijnden, E.: A patch that imparts unconditional stability to certain explicit integrators for SDEs, arXiv preprint arXiv:1010.4278 (2010)
Chen, X., Cruzeiro, A.B., Ratiu, T.S.: Constrained and stochastic variational principles for dissipative equations with advected quantities, arXiv preprint arXiv:1506.05024 (2015)
Clark, M.A., Kennedy, A.D.: Accelerating staggered-fermion dynamics with the rational hybrid Monte Carlo algorithm. Phys. Rev. D 75(1), 011502 (2007)
Clark, M.A., Kennedy, A.D., Silva, P.J.: Tuning HMC using Poisson brackets. arXiv preprint arXiv:0810.1315 (2008)
Cruzeiro, A.B., Holm, D.D., Ana Bela Cruzeiro: Momentum maps and stochastic clebsch action principles. Commun. Math. Phys. 357(2), 873–912 (2018)
Duane, Simon, Kennedy, A.D., Pendleton, B.J., Roweth, D.: Hybrid monte carlo. Phys. lett. B 195(2), 216–222 (1987)
Duncan, A.B., Lelievre, T., Pavliotis, G.A.: Variance reduction using nonreversible Langevin samplers. J. Stat. Phys. 163(3), 457–491 (2016)
Kennedy, A.D., Rossi, P.: Classical mechanics on group manifolds. Nucl. Phys. B 327, 782–790 (1989)
Kennedy, A.D., Silva, P.J., Clark, M.A.: Shadow Hamiltonians, poisson brackets, and gauge theories. Phys. Rev. D 87(3), 034511 (2013)
Mathias, Rousset, Gabriel, Stoltz, Tony, L.: Free Energy Computations: A Mathematical Perspective. World Scientific, Singapore (2010)
Neal, R.M., et al.: MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo 2(11), 2 (2011)
Ottobre, M.: Markov Chain Monte Carlo and irreversibility. Rep. Math. Phys. 77(3), 267–292 (2016)
Ottobre, M., Pillai, N.S., Pinski, F.J., Stuart, A.M., et al.: A function space HMC algorithm with second order Langevin diffusion limit. Bernoulli 22(1), 60–106 (2016)
Rey-Bellet, L., Spiliopoulos, K.: Irreversible langevin samplers and variance reduction: a large deviations approach. Nonlinearity 28(7), 2081 (2015)
Acknowledgements
The authors thank A. Duncan for the useful insights that helped improve this work.AA acknowledges EPSRC funding through award EP/N014529/1 via the EPSRCCentre for Mathematics of Precision Healthcare. ST acknowledges the Schrödinger scholarship scheme for funding during this work. AB was supported by a Roth Scholarship funded by the Department of Mathematics, Imperial College London, by EPSRC Fellowship (EP/J016934/3) and by The Alan Turing Institute under the EPSRC grant [EP/N510129/1].
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Arnaudon, A., Barp, A., Takao, S. (2019). Irreversible Langevin MCMC on Lie Groups. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_18
Download citation
DOI: https://doi.org/10.1007/978-3-030-26980-7_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-26979-1
Online ISBN: 978-3-030-26980-7
eBook Packages: Computer ScienceComputer Science (R0)