Numerical optimization of a sum-of-rank-1 decomposition for n-dimensional order-p symmetric tensors

In this paper, we present a sum-of-rank-1 type decomposition and its differential model for symmetric tensors and investigate the convergence properties of numerical gradient-based iterative optimization algorithms to obtain this decomposition. The decomposition we propose reinterprets the orthogonality property of the eigenvectors of symmetric matrices as a geometric constraint on the rank-1 matrix bases, which leads to a geometrically constrained eigenvector frame. Relaxing the orthogonality requirement, we developed a set of structured-bases that can be utilized to decompose any symmetric tensor into a similar constrained sum-of-rank-1 decomposition.