Abstract

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.

中文翻译：

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基于测地距离的梯度下降算法的应用

摘要

本文采用黎曼梯度算法和自然梯度算法来解决正定埃尔米特矩阵流形上的下降方向问题，其中以测地距离为目标函数。首先提出的问题是控制正定埃尔米特矩阵系统，其输出仅取决于其输入。测地距离被用作输出矩阵和目标矩阵的差。获得用于调节输入的控制器，使得输出矩阵尽可能接近目标矩阵。我们使用黎曼梯度算法显示流形上控制输入的轨迹。第二个应用是计算给定Toeplitz正定Hermitian矩阵的有限集的Karcher均值，定义为测地距离总和的最小化器。为了获得比传统算法更有效的迭代算法，提出了一种自然梯度算法来计算Karcher均值。提供了说明性仿真以显示所提出算法的计算行为。