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

We study the convergence of the Riemannian steepest descent algorithm on theGrassmann manifold for minimizing the block version of the Rayleigh quotient ofa symmetric matrix. Even though this problem is non-convex in the Euclideansense and only very locally convex in the Riemannian sense, we discover astructure for this problem that is similar to geodesic strong convexity,namely, weak-strong convexity. This allows us to apply similar arguments fromconvex optimization when studying the convergence of the steepest descentalgorithm but with initialization conditions that do not depend on the eigengap$\delta$. When $\delta>0$, we prove exponential convergence rates, whileotherwise the convergence is algebraic. Additionally, we prove that thisproblem is geodesically convex in a neighbourhood of the global minimizer ofradius $\mathcal{O}(\sqrt{\delta})$.

Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Riemannian Steepest Descent