Symmetric Tensor Nuclear Norms

TL;DR

This paper investigates methods for computing the nuclear norms of symmetric tensors, proposing Lasserre relaxations and analyzing their theoretical properties to enable practical computation of nuclear norms and decompositions.

Contribution

It introduces Lasserre relaxation techniques for symmetric tensor nuclear norm computation and studies their theoretical properties, extending methods to nonsymmetric tensors.

Findings

Lasserre relaxations effectively compute symmetric tensor nuclear norms.

Theoretical analysis confirms the validity of the relaxation methods.

Methods can be extended to nonsymmetric tensors.

Abstract

This paper studies nuclear norms of symmetric tensors. As recently shown by Friedland and Lim, the nuclear norm of a symmetric tensor can be achieved at a symmetric decomposition. We discuss how to compute symmetric tensor nuclear norms, depending on the tensor order and the ground field. Lasserre relaxations are proposed for the computation. The theoretical properties of the relaxations are studied. For symmetric tensors, we can compute their nuclear norms, as well as the nuclear decompositions. The proposed methods can be extended to nonsymmetric tensors.