License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ITCS.2018.5
URN: urn:nbn:de:0030-drops-83583
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8358/
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Balcan, Maria-Florina ; Liang, Yingyu ; Woodruff, David P. ; Zhang, Hongyang

Matrix Completion and Related Problems via Strong Duality

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Abstract

This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework and show that under certain dual conditions, the optimal solution of the matrix factorization program is the same as its bi-dual and thus the global optimality of the non-convex program can be achieved by solving its bi-dual which is convex. These dual conditions are satisfied by a wide class of matrix factorization problems, although matrix factorization problems are hard to solve in full generality. This analytical framework may be of independent interest to non-convex optimization more broadly.

We apply our framework to two prototypical matrix factorization problems: matrix completion and robust Principal Component Analysis (PCA). These are examples of efficiently recovering a hidden matrix given limited reliable observations of it. Our framework shows that exact recoverability and strong duality hold with nearly-optimal sample complexity guarantees for matrix completion and robust PCA.

BibTeX - Entry

@InProceedings{balcan_et_al:LIPIcs:2018:8358,
  author =	{Maria-Florina Balcan and Yingyu Liang and David P. Woodruff and Hongyang Zhang},
  title =	{{Matrix Completion and Related Problems via Strong Duality}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{5:1--5:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Anna R. Karlin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8358},
  URN =		{urn:nbn:de:0030-drops-83583},
  doi =		{10.4230/LIPIcs.ITCS.2018.5},
  annote =	{Keywords: Non-Convex Optimization, Strong Duality, Matrix Completion, Robust PCA, Sample Complexity}
}

Keywords: Non-Convex Optimization, Strong Duality, Matrix Completion, Robust PCA, Sample Complexity
Collection: 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)
Issue Date: 2018
Date of publication: 12.01.2018

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