License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.829
URN: urn:nbn:de:0030-drops-53394
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5339/
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Ge, Rong ; Ma, Tengyu

Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms

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Abstract

Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to n^{\lfloor p/2 \rceil} for a p-th order tensor. Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the first quasi-polynomial time algorithm that can decompose a random 3rd order tensor decomposition when the rank is as large as n^{3/2}/poly log n.

We also give a polynomial time algorithm for certifying the injective norm of random low rank tensors. Our tensor decomposition algorithm exploits the relationship between injective norm and the tensor components. The proof relies on interesting tools for decoupling random variables to prove better matrix concentration bounds.

BibTeX - Entry

@InProceedings{ge_et_al:LIPIcs:2015:5339,
  author =	{Rong Ge and Tengyu Ma},
  title =	{{Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{829--849},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5339},
  URN =		{urn:nbn:de:0030-drops-53394},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.829},
  annote =	{Keywords: sum of squares, overcomplete tensor decomposition}
}

Keywords: sum of squares, overcomplete tensor decomposition
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)
Issue Date: 2015
Date of publication: 13.08.2015

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