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Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space

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  • Lovric, Miroslav
  • Min-Oo, Maung
  • Ruh, Ernst A.

Abstract

The construction of a distance function between probability distributions is of importance in mathematical statistics and its applications. The distance function based on the Fisher information metric has been studied by a number of statisticians, especially in the case of the multivariate normal distribution (Gaussian) on n. It turns out that, except in the case n=1, where the Fisher metric describes the hyperbolic plane, it is difficult to obtain an exact formula for the distance function (although this can be achieved for special families with fixed mean or fixed covariance). We propose to study a slightly different metric on the space of multivariate normal distributions on n. Our metric is based on the fundamental idea of parametrizing this space as the Riemannian symmetric space SL(n+1)/SO(n+1). Symmetric spaces are well understood in Riemannian geometry, allowing us to compute distance functions and other relevant geometric data.

Suggested Citation

  • Lovric, Miroslav & Min-Oo, Maung & Ruh, Ernst A., 2000. "Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space," Journal of Multivariate Analysis, Elsevier, vol. 74(1), pages 36-48, July.
    • Handle: RePEc:eee:jmvana:v:74:y:2000:i:1:p:36-48
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    References listed on IDEAS

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    1. Burbea, Jacob & Rao, C. Radhakrishna, 1982. "Entropy differential metric, distance and divergence measures in probability spaces: A unified approach," Journal of Multivariate Analysis, Elsevier, vol. 12(4), pages 575-596, December.
    2. Calvo, Miquel & Oller, Josep M., 1990. "A distance between multivariate normal distributions based in an embedding into the siegel group," Journal of Multivariate Analysis, Elsevier, vol. 35(2), pages 223-242, November.
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    Cited by:

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    2. Branislav Popović & Marko Janev & Lidija Krstanović & Nikola Simić & Vlado Delić, 2022. "Measure of Similarity between GMMs Based on Geometry-Aware Dimensionality Reduction," Mathematics, MDPI, vol. 11(1), pages 1-22, December.
    3. Andai, Attila, 2009. "On the geometry of generalized Gaussian distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 777-793, April.
    4. Le Brigant, Alice & Puechmorel, Stéphane, 2019. "Quantization and clustering on Riemannian manifolds with an application to air traffic analysis," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 685-703.
    5. Branislav Popović & Lenka Cepova & Robert Cep & Marko Janev & Lidija Krstanović, 2021. "Measure of Similarity between GMMs by Embedding of the Parameter Space That Preserves KL Divergence," Mathematics, MDPI, vol. 9(9), pages 1-21, April.
    6. Lidija Krstanović & Branislav Popović & Marko Janev & Branko Brkljač, 2023. "Feature Map Regularized CycleGAN for Domain Transfer," Mathematics, MDPI, vol. 11(2), pages 1-21, January.
    7. Beatrice Acciaio & Anastasis Kratsios & Gudmund Pammer, 2022. "Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer," Papers 2201.13094, arXiv.org, revised Mar 2023.

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