Numerical Optimization for Symmetric Tensor Decomposition

TL;DR

This paper investigates numerical methods for decomposing real symmetric tensors into sums of outer products, focusing on real-valued and nonnegative decompositions, and demonstrates their effectiveness despite the nonconvex nature of the problem.

Contribution

It introduces straightforward numerical formulations for real symmetric tensor decomposition, including non-symmetric approaches, and evaluates their performance on test problems.

Findings

Proposed formulations are effective for low-rank symmetric tensor decomposition.

Ignoring symmetry can still yield successful decompositions.

Numerical results demonstrate the practicality of the methods despite nonconvexity.

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, 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.