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.
Distance is a fundamental concept when considering the information retrieval and cluster analysis of 3D information.
That is, a large number of information retrieval descriptor comparison and cluster analysis algorithms are built around
the very concept of the distance, such as the Mahalanobis or Manhattan distances, between points. Although not always
explicitly stated, a significant proportion of these distances are, by nature, Euclidian. This implies that it is assumed that
the data distribution, from a geometrical point of view, may be associated with a Euclidian flat space. In this paper, we
draw attention to the fact that this association is, in many situations, not appropriate. Rather, the data should often be
characterised by a Riemannian curved space. It is shown how to construct such a curved space and how to analyse its
geometry from a topological point of view. The paper also illustrates how, in curved space, the distance between two
points may be calculated. In addition, the consequences for information retrieval and cluster analysis algorithms are
discussed.
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