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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper, we study a general low-rank matrix recovery problem withlinear measurements corrupted by some noise. The objective is to understandunder what conditions on the restricted isometry property (RIP) of the problemlocal search methods can find the ground truth with a small error. By analyzingthe landscape of the non-convex problem, we first propose a global guarantee onthe maximum distance between an arbitrary local minimizer and the ground truthunder the assumption that the RIP constant is smaller than $1/2$. We show thatthis distance shrinks to zero as the intensity of the noise reduces. Our newguarantee is sharp in terms of the RIP constant and is much stronger than theexisting results. We then present a local guarantee for problems with anarbitrary RIP constant, which states that any local minimizer is eitherconsiderably close to the ground truth or far away from it. Next, we prove thestrict saddle property, which guarantees the global convergence of theperturbed gradient descent method in polynomial time. The developed resultsdemonstrate how the noise intensity and the RIP constant of the problem affectthe landscape of the problem.

Sharp Restricted Isometry Property Bounds for Low-rank Matrix Recovery Problems with Corrupted Measurements