# Numerical optimization for symmetric tensor decomposition

## Abstract

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. 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.

- Authors:

- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1497638

- Report Number(s):
- SAND2014-18405J

Journal ID: ISSN 0025-5610; 672377

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Mathematical Programming

- Additional Journal Information:
- Journal Volume: 151; Journal Issue: 1; Journal ID: ISSN 0025-5610

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Kolda, Tamara G. Numerical optimization for symmetric tensor decomposition. United States: N. p., 2015.
Web. doi:10.1007/s10107-015-0895-0.
```

```
Kolda, Tamara G. Numerical optimization for symmetric tensor decomposition. United States. doi:10.1007/s10107-015-0895-0.
```

```
Kolda, Tamara G. Sat .
"Numerical optimization for symmetric tensor decomposition". United States. doi:10.1007/s10107-015-0895-0. https://www.osti.gov/servlets/purl/1497638.
```

```
@article{osti_1497638,
```

title = {Numerical optimization for symmetric tensor decomposition},

author = {Kolda, Tamara G.},

abstractNote = {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. 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.},

doi = {10.1007/s10107-015-0895-0},

journal = {Mathematical Programming},

number = 1,

volume = 151,

place = {United States},

year = {2015},

month = {4}

}

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Cited by: 15 works

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#### Figures / Tables:

Table 1: Index and monomial representations for $\mathbb{S}^{[3,2]}$.

All figures and tables
(10 total)

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*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*