On the rank and the approximation of symmetric tensors

TL;DR

This paper explores various tensor ranks, introduces a new symmetric decomposable rank, and demonstrates its advantages in approximating symmetric tensors over traditional ranks.

Contribution

The paper introduces the symmetric decomposable rank and compares its effectiveness to tensor and nuclear ranks in symmetric tensor approximation.

Findings

Symmetric decomposable rank offers advantages in symmetric tensor approximation.

Using symmetric decomposable rank improves approximation quality.

The new rank concept simplifies symmetric tensor analysis.

Abstract

In this work we study different notions of ranks and approximation of tensors. We consider the tensor rank, the nuclear rank and we introduce the notion of symmetric decomposable rank, a notion of rank defined only on symmetric tensors. We show that when approximating symmetric tensors, using the symmetric decomposable rank has some significant advantages over the tensor rank and the nuclear rank.