Riemannian Gaussian distributions on the space of positive-de nite quaternion matrices

07/11/2017
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Recently, Riemannian Gaussian distributions were de ned on spaces of positive-de nite real and complex matrices. The present paper extends this de nition to the space of positive-de nite quaternion matrices. In order to do so, it develops the Riemannian geometry of the space of positive-de nite quaternion matrices, which is shown to be a Riemannian symmetric space of non-positive curvature. The paper gives original formulae for the Riemannian metric of this space, its geodesics, and distance function. Then, it develops the theory of Riemannian Gaussian distributions, including the exact expression of their probability density, their sampling algorithm and statistical inference.

Riemannian Gaussian distributions on the space of positive-denite quaternion matrices

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application/pdf Riemannian Gaussian distributions on the space of positive-de nite quaternion matrices Salem Said, Nicolas Le Bihan, Jonathan H. Manton
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contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit

Recently, Riemannian Gaussian distributions were de ned on spaces of positive-de nite real and complex matrices. The present paper extends this de nition to the space of positive-de nite quaternion matrices. In order to do so, it develops the Riemannian geometry of the space of positive-de nite quaternion matrices, which is shown to be a Riemannian symmetric space of non-positive curvature. The paper gives original formulae for the Riemannian metric of this space, its geodesics, and distance function. Then, it develops the theory of Riemannian Gaussian distributions, including the exact expression of their probability density, their sampling algorithm and statistical inference.
Riemannian Gaussian distributions on the space of positive-denite quaternion matrices
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Riemannian Gaussian distributions on the space of positive-definite quaternion matrices Salem Said1 , Nicolas Le Bihan2 , Jonathan H. Manton3 1. Laboratoire IMS (CNRS - UMR 5218), 2. Gipsa-lab (CNRS - UMR 5216), 3. The University of Melbourne, Dept. of Electrical and Electronic Engineering Abstract. Recently, Riemannian Gaussian distributions were defined on spaces of positive-definite real and complex matrices. The present paper extends this definition to the space of positive-definite quater- nion matrices. In order to do so, it develops the Riemannian geometry of the space of positive-definite quaternion matrices, which is shown to be a Riemannian symmetric space of non-positive curvature. The pa- per gives original formulae for the Riemannian metric of this space, its geodesics, and distance function. Then, it develops the theory of Rie- mannian Gaussian distributions, including the exact expression of their probability density, their sampling algorithm and statistical inference. Keywords: Riemannian Gaussian distribution, quaternion, positive-definite matrix, symplectic group, Riemannian barycentre 1 Introduction The Riemannian geometry of the spaces Pn and Hn , respectively of n × n positive-definite real and complex matrices, is well-known to the information science community [1, 2]. These spaces have the property of being Riemannian symmetric spaces of non-positive curvature [3, 4], Pn = GL(n, R)/O(n) Hn = GL(n, C)/U(n) where GL(n, R) and GL(n, C) denote the real and complex linear groups, and O(n) and U(n) the orthogonal and unitary groups. Using this property, Rieman- nian Gaussian distributions were recently introduced on Pn and Hn [5, 6]. The present paper introduces the Riemannian geometry of the space Qn of n × n positive-definite quaternion matrices, which is also a Riemannian symmetric space of non-positive curvature [4], Qn = GL(n, H)/Sp(n) where GL(n, H) denotes the quaternion linear group, and Sp(n) the compact symplectic group. It then studies Riemannian Gaussian distributions on Qn. The main results are the following : Proposition 1 gives the Riemannian metric of the space Qn, Proposition 2 expresses this metric in terms of polar coordinates on the space Qn, Proposition 3 uses Proposition 2 to compute the moment gener- ating function of a Riemannian Gaussian distribution on Qn, and Propositions 4 and 5 describe the sampling algorithm and maximum likelihood estimation of Riemannian Gaussian distributions on Qn. Motivation for studying matri- ces from Qn comes from their potential use in multidimensional bivariate signal processing [7]. 2 Quaternion matrices, GL(H) and Sp(n) Recall the non-commutative division algebra of quaternions, denoted H, is made up of elements q = q0 + q1 i+ q2 j+ q3 k where q0, q1, q2, q3 ∈ R, and the imaginary units i, j, k satisfy the relations [8] i2 = j2 = k2