Paper ID sheet UCLINMA2015.05
 Title

A Riemannian rankadaptive method for lowrank optimization
 Authors
 Guifang Zhou, Wen Huang, Kyle Gallivan, Paul Van Dooren, P.A. Absil
 Abstract

This paper presents an algorithm that solves optimization problems on a matrix manifold $\mathcal{M} \subseteq \mathbb{R}^{m \times
n}$ with an additional rank inequality constraint. The algorithm resorts to wellknown Riemannian optimization schemes on fixedrank manifolds, combined with new mechanisms to increase or decrease the rank. The convergence of the algorithm is analyzed and a weighted lowrank approximation problem is used to illustrate the efficiency and effectiveness of the algorithm.
 Key words
 lowrank optimization; rankconstrained optimization; Riemannian manifold; fixedrank manifold; lowrank approximation
 Status
 Accepted for publication in Neurocomputing, 2016
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 Errata

The errata refers to the publisher's version.
 Algorithm 3, in the "Require" line and line 1: replace "X_0" by "X_{1}". The sole purpose is to avoid the initial iterate being overwritten in the "for n" loop when n=0.
 Proof of Theorem 2, 10th line: replace "grad f(X_n)" by "grad f_r(X_n)".
 Proof of Theorem 2, displayed equation after "The contradiction argument is thus complete": replace the subscript "X_n" by "X_{n_j}".
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