An Efficient Gauss--Newton Algorithm for Symmetric Low-Rank Product Matrix Approximations

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dc.citation.firstpage1571en_US
dc.citation.issueNumber3en_US
dc.citation.journalTitleSIAM Journal on Optimizationen_US
dc.citation.lastpage1608en_US
dc.citation.volumeNumber25en_US
dc.contributor.authorLiu, Xinen_US
dc.contributor.authorWen, Zaiwenen_US
dc.contributor.authorZhang, Yinen_US
dc.date.accessioned2017-05-30T17:03:41Zen_US
dc.date.available2017-05-30T17:03:41Zen_US
dc.date.issued2015en_US
dc.description.abstractWe derive and study a Gauss--Newton method for computing a symmetric low-rank product $XX^{{T}}$, where $X \in{\mathbb{R}}^{n\times k}$ for $k<n$, that is the closest to a given symmetric matrix $A \in{\mathbb{R}}^{n\times n}$ in Frobenius norm. When $A=B^{{T}} B$ (or $BB^{{T}} $), this problem essentially reduces to finding a truncated singular value decomposition of $B$. Our Gauss--Newton method, which has a particularly simple form, shares the same order of iteration-complexity as a gradient method when $k \ll n$, but can be significantly faster on a wide range of problems. In this paper, we prove global convergence and a $Q$-linear convergence rate for this algorithm and perform numerical experiments on various test problems, including those from recently active areas of matrix completion and robust principal component analysis. Numerical results show that the proposed algorithm is capable of providing considerable speed advantages over Krylov subspace methods on suitable application problems where high-accuracy solutions are not required. Moreover, the algorithm possesses a higher degree of concurrency than Krylov subspace methods, thus offering better scalability on modern multi-/many-core computers.en_US
dc.identifier.citationLiu, Xin, Wen, Zaiwen and Zhang, Yin. "An Efficient Gauss--Newton Algorithm for Symmetric Low-Rank Product Matrix Approximations." <i>SIAM Journal on Optimization,</i> 25, no. 3 (2015) Society for Industrial and Applied Mathematics: 1571-1608. http://dx.doi.org/10.1137/140971464.en_US
dc.identifier.doihttp://dx.doi.org/10.1137/140971464en_US
dc.identifier.urihttps://hdl.handle.net/1911/94737en_US
dc.language.isoengen_US
dc.publisherSociety for Industrial and Applied Mathematicsen_US
dc.rightsArticle is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.en_US
dc.titleAn Efficient Gauss--Newton Algorithm for Symmetric Low-Rank Product Matrix Approximationsen_US
dc.typeJournal articleen_US
dc.type.dcmiTexten_US
dc.type.publicationpublisher versionen_US
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