On Uniqueness of Power Sum Decomposition

We propose an algorithm to compute power sum decompositions, which are motivated by applications in algebraic statistics. Power sum decomposition entails writing forms of degree d\cdotk as a sum of dth powers of k-forms. We show that under certain assumptions, the power sum problem for k-forms can be reduced to the classical case of power sums of linear forms. Semidefinite programming is used to perform this reduction. The semidefinite programming approach allows us to improve the currently best known rank bounds for the problem from m= \scrO(n/log(n)) to m= n-1, in a typical case. An implementation of the algorithm is provided. We complement the theoretical analysis with numerical experiments.