Alternate algorithms to most referenced techniques of numerical optimization to solve the symmetric rank-R approximation problem of symmetric tensors

Abstract

The tensor low-rank approximation and tensor CANDECOMP/PARAFAC (CP) decomposition are useful in various fields such as machine learning, dimension reduction, tensor completion, data visualization etc. A symmetric tensor is a higher order generalization of a symmetric matrix. Comon et al. (2008) show that every symmetric tensor has a symmetric CP-decomposition. In this paper, we study numerical methods of real-valued symmetric CP-decomposition of symmetric tensors. We present an alternate gradient descent method, an alternate BFGS method and an alternate Levenberg–Marquardt (L–M) method for real-valued symmetric rank-R approximation of symmetric tensors. Moreover, we prove the convergence and effectiveness of the algorithms. Numerical examples show that the alternate gradient descent method costs more computing time than the other two methods and the latter two methods have high success rate and good stability.