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

This work is concerned with the non-negative rank-1 robust principalcomponent analysis (RPCA), where the goal is to recover the dominantnon-negative principal components of a data matrix precisely, where a number ofmeasurements could be grossly corrupted with sparse and arbitrary large noise.Most of the known techniques for solving the RPCA rely on convex relaxationmethods by lifting the problem to a higher dimension, which significantlyincrease the number of variables. As an alternative, the well-knownBurer-Monteiro approach can be used to cast the RPCA as a non-convex andnon-smooth $\ell_1$ optimization problem with a significantly smaller number ofvariables. In this work, we show that the low-dimensional formulation of thesymmetric and asymmetric positive rank-1 RPCA based on the Burer-Monteiroapproach has benign landscape, i.e., 1) it does not have any spurious localsolution, 2) has a unique global solution, and 3) its unique global solutioncoincides with the true components. An implication of this result is thatsimple local search algorithms are guaranteed to achieve a zero globaloptimality gap when directly applied to the low-dimensional formulation.Furthermore, we provide strong deterministic and probabilistic guarantees forthe exact recovery of the true principal components. In particular, it is shownthat a constant fraction of the measurements could be grossly corrupted and yetthey would not create any spurious local solution.

Exact Guarantees on the Absence of Spurious Local Minima for Non-negative Rank-1 Robust Principal Component Analysis