Decomposing overcomplete 3rd order tensors using sum-of-squares algorithms

Conference Paper

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|p/2| for a p-th order tensor in Rnp. 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 n3/2/ 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.

Full Text

Duke Authors

Cited Authors

  • Ge, R; Ma, T

Published Date

  • August 1, 2015

Published In

Volume / Issue

  • 40 /

Start / End Page

  • 829 - 849

International Standard Serial Number (ISSN)

  • 1868-8969

International Standard Book Number 13 (ISBN-13)

  • 9783939897897

Digital Object Identifier (DOI)

  • 10.4230/LIPIcs.APPROX-RANDOM.2015.829

Citation Source

  • Scopus