Probability distributions from Riemannian geometry, generalized hybrid Monte Carlo sampling, and path integrals

E. Paquet; H. L. Viktor

doi:10.1117/12.872862

27 January 2011 Probability distributions from Riemannian geometry, generalized hybrid Monte Carlo sampling, and path integrals

E. Paquet, H. L. Viktor

Author Affiliations +

Proceedings Volume 7864, Three-Dimensional Imaging, Interaction, and Measurement; 78640X (2011) https://doi.org/10.1117/12.872862
Event: IS&T/SPIE Electronic Imaging, 2011, San Francisco Airport, California, United States

Abstract

When considering probabilistic pattern recognition methods, especially methods based on Bayesian analysis, the probabilistic distribution is of the utmost importance. However, despite the fact that the geometry associated with the probability distribution constitutes essential background information, it is often not ascertained. This paper discusses how the standard Euclidian geometry should be generalized to the Riemannian geometry when a curvature is observed in the distribution. To this end, the probability distribution is defined for curved geometry. In order to calculate the probability distribution, a Lagrangian and a Hamiltonian constructed from curvature invariants are associated with the Riemannian geometry and a generalized hybrid Monte Carlo sampling is introduced. Finally, we consider the calculation of the probability distribution and the expectation in Riemannian space with path integrals, which allows a direct extension of the concept of probability to curved space.

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E. Paquet and H. L. Viktor "Probability distributions from Riemannian geometry, generalized hybrid Monte Carlo sampling, and path integrals", Proc. SPIE 7864, Three-Dimensional Imaging, Interaction, and Measurement, 78640X (27 January 2011); https://doi.org/10.1117/12.872862

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