Semidefinite Relaxations for Best Rank-1 Tensor Approximations

TL;DR

This paper introduces semidefinite relaxation techniques based on sum of squares to efficiently find best rank-1 tensor approximations for symmetric and nonsymmetric tensors, demonstrating practical effectiveness through numerical experiments.

Contribution

It develops a novel semidefinite relaxation framework for tensor approximation problems, leveraging sum of squares representations and advanced optimization algorithms.

Findings

Semidefinite relaxations effectively approximate best rank-1 tensor solutions.

The approach scales to large problems using Newton-CG augmented Lagrangian methods.

Numerical experiments confirm the practicality and accuracy of the proposed methods.

Abstract

This paper studies the problem of finding best rank-1 approximations for both symmetric and nonsymmetric tensors. For symmetric tensors, this is equivalent to optimizing homogeneous polynomials over unit spheres; for nonsymmetric tensors, this is equivalent to optimizing multi-quadratic forms over multi-spheres. We propose semidefinite relaxations, based on sum of squares representations, to solve these polynomial optimization problems. Their properties and structures are studied. In applications, the resulting semidefinite programs are often large scale. The recent Newton-CG augmented Lagrangian method by Zhao, Sun and Toh is suitable for solving these semidefinite relaxations. Extensive numerical experiments are presented to show that this approach is practical in getting best rank-1 approximations.