Abstract
In this article we explore an algorithm for diffeomorphic random sampling of nonuniform probability distributions on Riemannian manifolds. The algorithm is based on optimal information transport (OIT)—an analogue of optimal mass transport (OMT). Our framework uses the deep geometric connections between the Fisher-Rao metric on the space of probability densities and the right-invariant information metric on the group of diffeomorphisms. The resulting sampling algorithm is a promising alternative to OMT, in particular as our formulation is semi-explicit, free of the nonlinear Monge–Ampere equation. Compared to Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when a large number of samples from a low dimensional nonuniform distribution is needed.
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References
Amari, S., Nagaoka, H.: Methods of information geometry. American Mathematical Society, Providence (2000)
Bauer, M., Bruveris, M., Michor, P.W.: Uniqueness of the Fisher-Rao metric on the space of smooth densities. Bull. Lond. Math. Soc. 48(3), 499–506 (2016)
Bauer, M., Joshi, S., Modin, K.: Diffeomorphic density matching by optimal information transport. SIAM J. Imaging Sci. 8(3), 1718–1751 (2015)
Friedrich, T.: Die Fisher-information und symplektische strukturen. Math. Nachr. 153(1), 273–296 (1991)
Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7(1), 65–222 (1982)
Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)
Khesin, B., Lenells, J., Misiołek, G., Preston, S.C.: Geometry of diffeomorphism groups, complete integrability and geometric statistics. Geom. Funct. Anal. 23(1), 334–366 (2013)
Khesin, B., Wendt, R.: The Geometry of Infinite-dimensional Groups. A Series of Modern Surveys in Mathematics, vol. 51. Springer, Berlin (2009)
Marzouk, Y., Moselhy, T., Parno, M., Spantini, A.: Sampling via measure transport: An introduction. In: Ghanem, R., Higdon, D., Owhadi, H. (eds.) Handbook of Uncertainty Quantification, pp. 1–14. Springer International Publishing, Cham (2016). doi:10.1007/978-3-319-11259-6_23-1
Miller, M.I., Trouvé, A., Younes, L.: On the metrics and euler-lagrange equations of computational anatomy. Annu. Rev. Biomed. Eng. 4, 375–405 (2002)
Modin, K.: Generalized Hunter-Saxton equations, optimal information transport, and factorization of diffeomorphisms. J. Geom. Anal. 25(2), 1306–1334 (2015)
Moselhy, T.A.E., Marzouk, Y.M.: Bayesian inference with optimal maps. J. Comput. Phys. 231(23), 7815–7850 (2012)
Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965)
Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eqn. 26(1–2), 101–174 (2001)
Reich, S.: A nonparametric ensemble transform method for Bayesian inference. SIAM J. Sci. Comput. 35(4), A2013–A2024 (2013)
Villani, C.: Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)
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Bauer, M., Joshi, S., Modin, K. (2017). Diffeomorphic Random Sampling Using Optimal Information Transport. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_16
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DOI: https://doi.org/10.1007/978-3-319-68445-1_16
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