Low Rank Symmetric Tensor Approximations

TL;DR

This paper introduces a novel method for approximating symmetric tensors with low rank tensors by leveraging generating polynomials and their approximate common zeros, providing quasi-optimal solutions when the original tensor is close to low rank.

Contribution

The paper proposes a new approach using generating polynomials to compute low rank symmetric tensor approximations, which is a significant advancement over existing methods.

Findings

The method effectively finds low rank approximations close to the original tensor.

The approach guarantees quasi-optimal solutions when the tensor is near a low rank tensor.

Experimental results demonstrate the efficiency and accuracy of the proposed method.

Abstract

For a given symmetric tensor, we aim at finding a new one whose symmetric rank is small and that is close to the given one. There exist linear relations among the entries of low rank symmetric tensors. Such linear relations can be expressed by polynomials, which are called generating polynomials. We propose a new approach for computing low rank approximations by using generating polynomials. First, we estimate a set of generating polynomials that are approximately satisfied by the given tensor. Second, we find approximate common zeros of these polynomials. Third, we use these zeros to construct low rank tensor approximations. If the symmetric tensor to be approximated is sufficiently close to a low rank one, we show that the computed low rank approximations are quasi-optimal.