A0308
Title: Bi-invariant dissimilarity measures for sample distributions in lie groups
Authors: Christoph von Tycowicz - Zuse Institute Berlin (Germany) [presenting]
Hans-Christian Hege - Zuse Institute Berlin (Germany)
Martin Hanik - Technical University Berlin (Germany)
Abstract: Data sets sampled in Lie groups are widespread, and as with multivariate data, it is important for many applications to assess the differences between the sets in terms of their distributions. Indices for this task are usually derived by considering the Lie group as a Riemannian manifold. Then, however, compatibility with the group operation is guaranteed only if a bi-invariant metric exists, which is not the case for most non-compact and non-commutative groups. By using an affine connection structure instead, we will obtain bi-invariant generalizations of well-known dissimilarity measures: the Hotelling $T^2$ statistic, the Bhattacharyya distance, and the Hellinger distance. Each of the dissimilarity measures matches its multivariate counterpart for Euclidean data and is translation-invariant, so that biases, e.g., through an arbitrary choice of reference, are avoided. We will examine the potential of these dissimilarity measures by performing group tests on knee configurations and epidemiological shape data.
