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In this paper, we present a method to certify the approximation quality of alow rank tensor to a given third order symmetric tensor. Under mildassumptions, best low rank approximation is attained if a control parameter iszero or quantified quasi-optimal low rank approximation is obtained if thecontrol parameter is positive.This is based on a primal-dual method forcomputing a low rank approximation for a given tensor. The certification isderived from the global optimality of the primal and dual problems, and ischaracterized by easily checkable relations between the primal and the dualsolutions together with another rank condition. The theory is verifiedtheoretically for orthogonally decomposable tensors as well as numericallythrough examples in the general case.

Quantifying low rank approximations of third order symmetric tensors