Geodesic Monte Carlo on embedded manifolds

Scandinavian Journal of Statistics

Volumn 40, Issue 4, 2013, Pages 825-845

  Byrne, Simon ^a   Girolami, Mark ^a  

^a UNIVERSITY COLLEGE LONDON   (United Kingdom)

Author keywords

Directional statistics; Geodesic, Hamiltonian Monte Carlo; Riemannian manifold; Stiefel manifold

Indexed keywords

EID: 84888128895     PISSN: 03036898     EISSN: 14679469     Source Type: Journal    
DOI: 10.1111/sjos.12036     Document Type: Article

References (33)

1. Abraham, R. & Marsden, J.E. (1978). Foundations of mechanics, (2nded.)., Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass. ISBN: 0-8053-0102-X.
2. Amari, S. & Nagaoka, H. (2000). Methods of information geometry, Translations of Mathematical Monographs, vol. 191, American Mathematical Society, Providence, RI. ISBN: 0-8218-0531-2.
3. Beskos, A., Pinski, F.J., Sanz-Serna, J.M. & Stuart, A.M. (2011). Hybrid Monte Carlo on Hilbert spaces. Stochastic Process. Appl. 121, (10), 2201-2230, DOI 10.1016/j.spa.2011.06.003.
4. Brubaker, M., Salzmann, M. & Urtasun, R. (2012). A family of MCMC methods on implicitly defined manifolds. In JMLR Workshop and Conference Proceedings, Vol.22; 161-172. Available on http://jmlr.csail.mit.edu/proceedings/papers/v22/brubaker12/brubaker12. pdf.
5. Cardoso, J.R. & Leite, F.S. (2010). Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. J. Comput. Appl. Math. 233, (11), 2867-2875. DOI: 10.1016/j.cam.2009.11.032.
6. Diaconis, P., Holmes, S. & Shahshahani, M. (2013). Sampling from a manifold. In Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton (eds G. Jones & X. Shen ), Institute of Mathematical Statistics.
7. do Carmo, M.P. (1976). Differential geometry of curves and surfaces, Prentice-Hall Inc., Englewood Cliffs, N.J.
8. do Carmo, M.P. (1992). Riemannian geometry, Mathematics: theory & applications, Birkhäuser Boston Inc., Boston, MA. ISBN: 0-8176-3490-8.
9. Dobigeon, N. & Tourneret, J.-Y. (2010). Bayesian orthogonal component analysis for sparse representation. IEEE Trans. Signal Process. 58, (5), 2675-2685. ISSN: 1053-587X, DOI: 10.1109/TSP.2010.2041594, Available on http://dx.doi.org/10.1109/TSP.2010.2041594.
10. Duane, S., Kennedy, A.D., Pendleton, B.J. & Roweth, D. (1987). Hybrid Monte Carlo. Phys. Lett. B 195, 216-222.
11. Edelman, A., Arias, T.A. & Smith, S.T. (1999). The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20, (2), 303-353. DOI: 10.1137/S0895479895290954.
12. Federer, H. (1969). Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York.
13. Gallier, J. & Xu, D. (2002). Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. Int. J. Rob. Autom. 17, (4), 1-11.
14. Geyer, C.J. (1991). Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics: The 23rd Symposium on the Interface (ed Keramigas, E.), Interface Foundation, Fairfax; 156-163.
15. Gilks, W.R., Richardson, S. & Spiegelhalter, D.J. (eds). (1996). Markov chain Monte Carlo in practice, Interdisciplinary Statistics, Chapman & Hall, London. ISBN: 0-412-05551-1.
16. Girolami, M. & Calderhead, B. (2011). Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 73, (2), 123-214. With discussion and a reply by the authors, DOI: 10.1111/j.1467-9868.2010.00765.x.
17. Hairer, E., Lubich, C. & Wanner, G. (2006). Geometric numerical integration, (2nded.)., Springer Series in Computational Mathematics, vol.31, Springer-Verlag, Berlin. Structure-preserving algorithms for ordinary differential equations, ISBN: 3-540-30663-3; 978-3-540-30663-4.
18. Hankin, R.K.S. (2010). A generalization of the Dirichlet distribution. J. Stat. Softw. 33, (11), 1-18. Available on http://www.jstatsoft.org/v33/i11.
19. Hoff, P.D. (2009). Simulation of the matrix Bingham-von Mises-Fisher distribution, with applications to multivariate and relational data. J. Comput. Graph. Statist. 18, (2), 438-456. DOI: 10.1198/jcgs.2009.07177.
20. Jolliffe, I.T. (1986). Principal component analysis, Springer Series in Statistics, Springer-Verlag, New York. ISBN: 0-387-96269-7.
21. Konukoglu, E., Relan, J., Cilingir, U., Menze, B.H., Chinchapatnam P., Jadidi, A., Cochet, H., Hocini, M., Delingette, H., Jaïs, P., Haïssaguerre, M., Ayache, N. & Sermesant, M. (2011). Efficient probabilistic model personalization integrating uncertainty on data and parameters: application to eikonal-diffusion models in cardiac electrophysiology. Prog. Biophys. Mol. Biol. 107, (1), 134-146.
22. Le Polain de Waroux, O., Maguire, H. & Moren, A. (2012). The case-cohort design in outbreak investigations. Euro Surveillance: Bulletin Europeen sur les Maladies Transmissibles 17, (25), 11-15.
23. Liu, J.S. (2008). Monte Carlo strategies in scientific computing, Springer Series in Statistics, Springer, New York. pp. xvi+343. isbn: 978-0-387-76369-9; 0-387-95230-6.
24. Mardia, K.V. & Jupp, P.E. (2000). Directional statistics, Wiley Series in Probability and Statistics, John Wiley & Sons Ltd., Chichester. ISBN: 0-471-95333-4.
25. Martin, J., Wilcox, L.C., Burstedde, C. & Ghattas, O. (2012). A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion. SIAM J. Sci. Comput. 34, (3), A1460-A1487.
26. Morgan, F. (2009). Geometric measure theory, (4thed.)., Elsevier/Academic Press, Amsterdam. A beginner's guide, ISBN: 978-0-12-374444-9.
27. Nash, J. (1956). The imbedding problem for Riemannian manifolds. Ann. of Math. (2) 63, 20-63.
28. Neal, R.M. (2011). MCMC using Hamiltonian dynamics. In Handbook of Markov chain Monte Carlo, Chapman & Hall/CRC Handb. Mod. Stat. Methods CRC Press, Boca Raton, FL; 113-162.
29. Raue, A., Kreutz, C., Theis, F.J. & Timmer, J. (2012). Joining forces of Bayesian and frequentist methodology: a study for inference in the presence of non-identifiability. Phil. Trans. R. Soc. A 371, (1984).
30. Shahbaba, B., Lan, S., Johnson, W.O. & Neal, R.M. (2011). Split Hamiltonian Monte Carlo. arXiv: 1106.5941.
31. Singh, H., Hnizdo, V. & Demchuk, E. (2002). Probabilistic model for two dependent circular variables. Biometrika 89, (3), 719-723. DOI: 10.1093/biomet/89.3.719.
32. Stan Development Team. (2012). Stan: a C++ library for probability and sampling, version 1.0. Available on http://mc-stan.org/.
33. Vanlier, J., Tiemann, C.A., Hilbers, P.A.J. & van Riel, N.A.W. (2012). An integrated strategy for prediction uncertainty analysis. Bioinformatics 28, (8), 1130-1135.