Geometric distance between positive definite matrices of different dimensions

2018/06/04 by Lim, Lek-Heng, Sepulchre, Rodolphe, Ye, Ke · 2 citations
#15A18 #15B48 #51K99 #53C25 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1806.01428

Abstract

We show how the Riemannian distance on \mathbbSⁿ₊₊, the cone of n× n real symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different dimensions. Given that \mathbbSⁿ₊₊ also parameterizes n-dimensional ellipsoids, and inner products on ℝⁿ, n × n covariance matrices of nondegenerate probability distributions, this gives us a natural way to define a geometric distance between a pair of such objects of different dimensions.

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