Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space

By Miroslav Lovric, Maung Min-Oo, E. Ruh. 2000

101 citations recorded by the source corpus.

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 Rn. 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 Rn. 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.

Subfields: Artificial Intelligence, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b88111dc2f1f264379e4e602d1c1492c8f448705

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Citations in the corpus

Distribution Calibration in Riemannian Symmetric Space (2011, unknown): “According to [20], three consecutive projections, i.”
Hierarchical Gaussian Descriptor for Person Re-identification (2016, methodology): “From an analysis in the information geometry literature [25], the space of d-dimensional multivariate Gaussians can be embedded into the d + 1 dimensional SPD matrices space denoted by Sym+d+1.”
Cramer-Rao Lower Bound and Information Geometry (2013, methodology): “For univariate normal distributions, Rao’s distance amounts to computing the hyperbolic distance for H( 1 √ 2 ), see [34].”
Log-Euclidean Metric Learning on Symmetric Positive Definite Manifold with Application to Image Set Classification (2015, methodology): “Following our previous works (Huang et al., 2015b; Wang et al., 2015) which employ information geometry theory (Amari & Nagaoka, 2000; Lovrić et al., 2000), we also transform the Gaussian modelN (m,S) into a (d+1)- dimensional SPD matrix as |S|− 1 d+1 [ S +mmT m mT 1 ] .”