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

Incorporating a non-Euclidean variable metric to first-order algorithms isknown to bring enhancement. However, due to the lack of an optimal choice, suchan enhancement appears significantly underestimated. In this work, we establisha metric selection principle via optimizing a convergence rate upper-bound. Forgeneral l1-minimization, we propose an optimal metric choice with closed-formexpressions guaranteed. Equipping such a variable metric, we prove that theoptimal solution to the l1 problem will be obtained via a one-time proximaloperator evaluation. Our technique applies to a large class of fixed-pointalgorithms, particularly the ADMM, which is popular, general, and requiresminimum assumptions.  The key to our success is the employment of an unscaled/equilibrateupper-bound. We show that there exists an implicit scaling that poses a hiddenobstacle to optimizing parameters. This turns out to be a fundamental issueinduced by the classical parametrization. We note that the conventional wayalways associates the parameter to the range of a function/operator. This turnsout not a natural way, causing certain symmetry losses, definitioninconsistencies, and unnecessary complications, with the well-known Moreauidentity being the best example. We propose equilibrate parametrization, whichassociates the parameter to the domain of a function, and to both the domainand range of a monotone operator. A series of powerful results are obtainedowing to the new parametrization. Quite remarkably, the preconditioningtechnique can be shown as equivalent to the metric selection issue.

Equilibrate Parametrization: Optimal Metric Selection with Provable One-iteration Convergence for $ l_1 $-minimization