Application of gradient descent algorithms based on geodesic distances  

作　　者：Xiaomin DUAN Huafei SUN Linyu PENG 

机构地区：[1]School of Science,Dalian Jiaotong University,Dalian 116028,China [2]School of Mathematics and Statistics,Beijing Institute of Technology,Beijing 100081,China [3]Waseda Institute for Advanced Study,Waseda University,Tokyo 169-8050,Japan

出　　处：《Science China(Information Sciences)》2020年第5期172-182,共11页中国科学（信息科学）（英文版）

基　　金：National Science and Technology Major Project of China(Grant No.2016YFF02030012);National Natural Science Foundation of China(Grant No.61401058);partially supported by National Natural Science Foundation of China(Grant No.61179031);Natural Science Foundation of Liaoning Province(Grant No.20180550112);supported by JSPS Grant-in-Aid for Scientific Research(Grant No.16KT0024);the MEXT“Top Global University Project”;Waseda University Grant for Special Research Projects(Grant Nos.2019C-179,2019E-036);Waseda University Grant Program for Promotion of International Joint Research。

摘　　要：In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the objective function. The first proposed problem is the control for positive definite Hermitian matrix systems whose outputs only depend on their inputs. The geodesic distance is adopted as the difference of the output matrix and the target matrix. The controller to adjust the input is obtained such that the output matrix is as close as possible to the target matrix. We show the trajectory of the control input on the manifold using the Riemannian gradient algorithm. The second application is to compute the Karcher mean of a finite set of given Toeplitz positive definite Hermitian matrices, which is defined as the minimizer of the sum of geodesic distances. To obtain more efficient iterative algorithm than traditional ones, a natural gradient algorithm is proposed to compute the Karcher mean. Illustrative simulations are provided to show the computational behavior of the proposed algorithms.

关 键 词：system control RIEMANNIAN GRADIENT ALGORITHM natural GRADIENT ALGORITHM Karcher mean TOEPLITZ POSITIVE definite HERMITIAN matrix 

分 类 号：TP301.6[自动化与计算机技术—计算机系统结构]