Bi-invariant Dissimilarity Measures for Sample Distributions in Lie Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910337427570688 |
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| author | Hanik, Martin Hege, Hans-Christian von Tycowicz, Christoph |
| author_facet | Hanik, Martin Hege, Hans-Christian von Tycowicz, Christoph |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12901 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bi-invariant Dissimilarity Measures for Sample Distributions in Lie Groups Hanik, Martin Hege, Hans-Christian von Tycowicz, Christoph Methodology Statistics Theory 62R30 22E70 62R30, 22E70, 53Z50 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. |
| title | Bi-invariant Dissimilarity Measures for Sample Distributions in Lie Groups |
| topic | Methodology Statistics Theory 62R30 22E70 62R30, 22E70, 53Z50 |
| url | https://arxiv.org/abs/2402.12901 |