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

Nonsmooth composite optimization with orthogonality constraints has a broadspectrum of applications in statistical learning and data science. However,this problem is generally challenging to solve due to its non-convex andnon-smooth nature. Existing solutions are limited by one or more of thefollowing restrictions: (i) they are full gradient methods that require highcomputational costs in each iteration; (ii) they are not capable of solvinggeneral nonsmooth composite problems; (iii) they are infeasible methods and canonly achieve the feasibility of the solution at the limit point; (iv) they lackrigorous convergence guarantees; (v) they only obtain weak optimality ofcritical points. In this paper, we propose \textit{\textbf{OBCD}}, a new BlockCoordinate Descent method for solving general nonsmooth composite problemsunder Orthogonality constraints. \textit{\textbf{OBCD}} is a feasible methodwith low computation complexity footprints. In each iteration, our algorithmupdates $k$ rows of the solution matrix ($k\geq2$ is a parameter) to preservethe constraints. Then, it solves a small-sized nonsmooth composite optimizationproblem under orthogonality constraints either exactly or approximately. Wedemonstrate that any exact block-$k$ stationary point is always an approximateblock-$k$ stationary point, which is equivalent to the critical stationarypoint. We are particularly interested in the case where $k=2$ as the resultingsubproblem reduces to a one-dimensional nonconvex problem. We propose abreakpoint searching method and a fifth-order iterative method to solve thisproblem efficiently and effectively. We also propose two novel greedystrategies to find a good working set to further accelerate the convergence of\textit{\textbf{OBCD}}. Finally, we have conducted extensive experiments onseveral tasks to demonstrate the superiority of our approach.

A Block Coordinate Descent Method for Nonsmooth Composite Optimization under Orthogonality Constraints