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2021 | OriginalPaper | Buchkapitel

Gaussian Distributions on Riemannian Symmetric Spaces in the Large N Limit

verfasst von : Simon Heuveline, Salem Said, Cyrus Mostajeran

Erschienen in: Geometric Science of Information

Verlag: Springer International Publishing

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Abstract

We consider the challenging problem of computing normalization factors of Gaussian distributions on certain Riemannian symmetric spaces. In some cases, such as the space of Hermitian positive definite matrices or hyperbolic space, it is possible to compute them exactly using techniques from random matrix theory. However, in most cases which are important to applications, such as the space of symmetric positive definite (SPD) matrices or the Siegel domain, this is only possible numerically. Moreover, when we consider, for instance, high-dimensional SPD matrices, the known algorithms can become exceedingly slow. Motivated by notions from theoretical physics, we will discuss how to approximate these normalization factors in the large N limit: an approximation that gets increasingly better as the dimension of the underlying symmetric space (more precisely, its rank) gets larger. We will give formulas for leading order terms in the case of SPD matrices and related spaces. Furthermore, we will characterize the large N limit of the Siegel domain through a singular integral equation arising as a saddle-point equation.

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Metadaten
Titel
Gaussian Distributions on Riemannian Symmetric Spaces in the Large N Limit
verfasst von
Simon Heuveline
Salem Said
Cyrus Mostajeran
Copyright-Jahr
2021
DOI
https://doi.org/10.1007/978-3-030-80209-7_3

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