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Bi-invariant Dissimilarity Measures for Sample Distributions in Lie Groups

Martin Hanik, Hans-Christian Hege, Christoph von Tycowicz

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. We show here that if one considers an affine connection structure instead, one obtains bi-invariant generalizations of well-known dissimilarity measures: a Hotelling $T^2$ statistic, Bhattacharyya distance and 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 further derive non-parametric two-sample tests that are bi-invariant and consistent. We demonstrate the potential of these dissimilarity measures by performing group tests on data of knee configurations and epidemiological shape data. Significant differences are revealed in both cases.

Bi-invariant Dissimilarity Measures for Sample Distributions in Lie Groups

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. We show here that if one considers an affine connection structure instead, one obtains bi-invariant generalizations of well-known dissimilarity measures: a Hotelling statistic, Bhattacharyya distance and 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 further derive non-parametric two-sample tests that are bi-invariant and consistent. We demonstrate the potential of these dissimilarity measures by performing group tests on data of knee configurations and epidemiological shape data. Significant differences are revealed in both cases.
Paper Structure (23 sections, 10 theorems, 67 equations, 3 figures)

This paper contains 23 sections, 10 theorems, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Affine Manifolds and Lie Groups
  4. The Group Mean
  5. Sample Covariance and Bi-invariant Mahalanobis Distance
  6. Bi-invariant Dissimilarity Measures for Sample Distributions on Lie Groups
  7. Extending the Hotelling T Squared Statistic
  8. Extending the Bhattacharyya Distance
  9. Connection of the Bi-invariant Bhattacharyya Distance to Densities
  10. Experiments
  11. Configurations of the Knee Joint under Osteoarthritis
  12. Data Description
  13. Encoding Relative Positions in SE(3)
  14. Bi-invariant Permutation Test
  15. Hippocampi Shapes under Alzheimer's
  16. ...and 8 more sections

Key Result

Theorem 2.2

\newlabelthm:mean_equivariance0 Let $G$ be a Lie group endowed with its CCS connection and $\overline{g}$ be a group mean of $g_1,\dots,g_m \in V \subseteq G$. Then, for any $f \in G$, the group means of the left translated data $(fg_1,\dots,fg_m)$, right translated data $(g_1f,\dots,g_mf)$ and in

Figures (3)

  • Figure 1: Visualization of the knee bones and the computed orthonormal frames. On the left, the distal femur (top) and the proximal tibia (below) are shown; the meshes (and their relative position) were reconstructed from OA data. The space between both (i.e., the joint space) is filled with cartilage, ligaments, and menisci (all not shown). On the right, we depict the orthonormal frames that correspond to both meshes.
  • Figure 2: Group means of right hippocampi for cognitive normal (orange, transparent) and impaired (grey) subjects overlaid.
  • Figure 3: Group tests for differences in the distribution of right hippocampi for cognitive normal and impaired subjects. Results for the bi-invariant Hotelling $T^2$ statistic are shown at the top and for the Bhattacharyya distance at the bottom. Each triangle of the CN mean is color coded according to its $p$-value (FDR corrected) using the colormap 0.0 0.05.

Theorems & Definitions (27)

  • Definition 2.1: Group mean
  • Theorem 2.2: Equivariance of the group mean
  • Definition 3.1: Centralized sample covariance
  • Definition 3.2: Pooled sample covariance
  • Lemma 3.3
  • Proof 1
  • Corollary 3.4
  • Definition 3.5: Bi-invariant Hotelling $T^2$ statistic
  • Theorem 3.6: Properties of the bi-invariant Hotelling $T^2$ statistic
  • Proof 2
  • ...and 17 more