English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, the Riemannian gradient algorithm and the natural gradientalgorithm are applied to solve descent direction problems on the manifold ofpositive definite Hermitian matrices, where the geodesic distance is consideredas the cost function. The first proposed problem is control for positivedefinite Hermitian matrix systems whose outputs only depend on their inputs.The geodesic distance is adopted as the difference of the output matrix and thetarget matrix. The controller to adjust the input is obtained such that theoutput matrix is as close as possible to the target matrix. We show thetrajectory of the control input on the manifold using the Riemannian gradientalgorithm. The second application is to compute the Karcher mean of a finiteset of given Toeplitz positive definite Hermitian matrices, which is defined asthe minimizer of the sum of geodesic distances. To obtain more efficientiterative algorithm compared with traditional ones, a natural gradientalgorithm is proposed to compute the Karcher mean. Illustrative simulations areprovided to show the computational behavior of the proposed algorithms.

Application of gradient descent algorithms based on geodesic distances