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dc.contributor.authorWei, Ke
dc.contributor.authorCai, Jian Feng
dc.contributor.authorChan, Tony Fan-Cheong
dc.contributor.authorLeung, Shingyu
dc.date.accessioned2020-03-01T07:05:07Z
dc.date.available2020-03-01T07:05:07Z
dc.date.issued2020-02-14
dc.date.submitted2019-01
dc.identifier.citationWei, K., Cai, J.-F., F. Chan, T., & Leung, S. (2020). Guarantees of riemannian optimization for low rank matrix completion. Inverse Problems & Imaging, 14(2), 233–265. doi:10.3934/ipi.2020011
dc.identifier.doi10.3934/ipi.2020011
dc.identifier.urihttp://hdl.handle.net/10754/661816
dc.description.abstractWe establish the exact recovery guarantees for a class of Riemannian optimization methods based on the embedded manifold of low rank matrices for matrix completion. Assume m entries of an n×n rank r matrix are sampled independently and uniformly with replacement. We first show that with high probability the Riemannian gradient descent and conjugate gradient descent algorithms initialized by one step hard thresholding are guaranteed to converge linearly to the measured matrix provided m ≥ Cκn1.5r log1.5(n), where Cκ is a numerical constant depending on the condition number of the measured matrix. Then the sampling complexity is further improved to m ≥ Cκnr2 log2(n) via the resampled Riemannian gradient descent initialization. The analysis of the new initialization procedure relies on an asymmetric restricted isometry property of the sampling operator and the curvature of the low rank matrix manifold. Numerical simulation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements.
dc.description.sponsorshipThe first author is supported by National Science Foundation of China (NSFC) 11801088 and Shanghai Sailing Program 18YF1401600. The second author is supported by Hong Kong Research Grant Council (HKRGC) General Research Fund (GRF) 16306317.
dc.publisherAmerican Institute of Mathematical Sciences (AIMS)
dc.relation.urlhttp://aimsciences.org//article/doi/10.3934/ipi.2020011
dc.relation.urlhttps://www.aimsciences.org/article/exportPdf?id=ee28e854-8cf5-44bf-b7db-8dbb354f806f
dc.relation.urlhttps://arxiv.org/pdf/1603.06610.pdf
dc.rightsThis is a pre-copy-editing, author-produced PDF of an article accepted for publication in [JournalTitle] following peer review. The definitive publisher-authenticated version is available online at: http://doi.org/10.3934/ipi.2020011
dc.rightsThis file is an open access version redistributed from: https://arxiv.org/pdf/1603.06610.pdf
dc.titleGuarantees of Riemannian optimization for low rank matrix completion
dc.typeArticle
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.contributor.departmentOffice of the President
dc.identifier.journalInverse Problems and Imaging
dc.rights.embargodate2021-02-14
dc.eprint.versionPost-print
dc.contributor.institutionSchool of Data Science, Fudan University, Shanghai, China
dc.contributor.institutionDepartment of Mathematics Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China
dc.identifier.arxivid1603.06610
kaust.personChan, Tony Fan-Cheong
dc.date.accepted2019-10
refterms.dateFOA2020-09-21T11:42:42Z
dc.date.published-online2020-02-14
dc.date.published-print2020


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