{"title":"Riemannian Lp Center of Mass for Scatter Matrix Estimation in Complex Elliptically Symmetric Distributions","authors":"Mengjiao Tang, Yao Rong, Chen Chen","doi":"10.23919/fusion49465.2021.9626967","DOIUrl":null,"url":null,"abstract":"A popular method for fusing a set of covariance matrix estimates (with unavailable correlation) is to solve their geometrical mean or median, which is defined by a Riemannian geometry of Hermitian positive-definite (HPD) matrices. The most well-known such geometry is identical to the Fisher information geometry of multivariate Gaussian distributions with a fixed mean. This paper identifies the space of HPD matrices with the manifold of centered (i.e., zero-mean) complex elliptically symmetric (CES) distributions. First, the Fisher information matrix for the CES distributions defines a different Riemannian metric on HPD matrices, and the induced Riemannian geometry is studied. Then, the Riemannian Lp mean of some HPD matrices is calculated to produce a final estimation for the scatter matrix (proportional to the covariance matrix) of a CES distribution. While the corresponding objective function is proven to be gconvex, a Riemannian gradient descent algorithm is given to compute the solution. Finally, numerical examples are provided to illustrate the derived geometrical structure and its application to target detection.","PeriodicalId":226850,"journal":{"name":"2021 IEEE 24th International Conference on Information Fusion (FUSION)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"2021 IEEE 24th International Conference on Information Fusion (FUSION)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.23919/fusion49465.2021.9626967","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
A popular method for fusing a set of covariance matrix estimates (with unavailable correlation) is to solve their geometrical mean or median, which is defined by a Riemannian geometry of Hermitian positive-definite (HPD) matrices. The most well-known such geometry is identical to the Fisher information geometry of multivariate Gaussian distributions with a fixed mean. This paper identifies the space of HPD matrices with the manifold of centered (i.e., zero-mean) complex elliptically symmetric (CES) distributions. First, the Fisher information matrix for the CES distributions defines a different Riemannian metric on HPD matrices, and the induced Riemannian geometry is studied. Then, the Riemannian Lp mean of some HPD matrices is calculated to produce a final estimation for the scatter matrix (proportional to the covariance matrix) of a CES distribution. While the corresponding objective function is proven to be gconvex, a Riemannian gradient descent algorithm is given to compute the solution. Finally, numerical examples are provided to illustrate the derived geometrical structure and its application to target detection.