Mathematical Programming

, Volume 151, Issue 1, pp 225–248 | Cite as

Numerical optimization for symmetric tensor decomposition

  • Tamara G. Kolda
Full Length Paper Series B


We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on real-valued decompositions, both unconstrained and nonnegative, for problems with low-rank structure. We discuss when solutions exist and how to formulate the mathematical program. Numerical results show the properties of the proposed formulations (including one that ignores symmetry) on a set of test problems and illustrate that these straightforward formulations can be effective even though the problem is nonconvex.


Symmetric Outer product Canonical polyadic Tensor decomposition Completely positive Nonnegative 

Mathematics Subject Classification

90C30 15A69 



The anonymous referees provided extremely useful feedback that has greatly improved the manuscript. This material is based upon work supported by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energy’s National Nuclear Security Administration under contract DE–AC04–94AL85000.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  1. 1.Sandia National LaboratoriesLivermoreUSA

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