Quantifying low rank approximations of third order symmetric tensors

TL;DR

This paper introduces a primal-dual method to certify the quality of low rank approximations for third order symmetric tensors, providing theoretical guarantees and practical verification techniques.

Contribution

It presents a novel certification method for low rank tensor approximations based on primal-dual optimality conditions, applicable to symmetric tensors.

Findings

Certification method guarantees approximation quality under mild conditions

Theoretical validation for orthogonally decomposable tensors

Numerical experiments confirm effectiveness in general cases

Abstract

In this paper, we present a method to certify the approximation quality of a low rank tensor to a given third order symmetric tensor. Under mild assumptions, best low rank approximation is attained if a control parameter is zero or quantified quasi-optimal low rank approximation is obtained if the control parameter is positive.This is based on a primal-dual method for computing a low rank approximation for a given tensor. The certification is derived from the global optimality of the primal and dual problems, and is characterized by easily checkable relations between the primal and the dual solutions together with another rank condition. The theory is verified theoretically for orthogonally decomposable tensors as well as numerically through examples in the general case.