Application of gradient descent algorithms based on geodesic distances

Xiaomin Duan, Huafei Sun, Linyu Peng

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number152201
JournalScience China Information Sciences
Volume63
Issue number5
DOIs
Publication statusPublished - 2020 May 1
Externally publishedYes

Keywords

  • Karcher mean
  • natural gradient algorithm
  • Riemannian gradient algorithm
  • system control
  • Toeplitz positive definite Hermitian matrix

ASJC Scopus subject areas

  • Computer Science(all)

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