Unique powers-of-forms decompositions from simple Gram spectrahedra

Abstract: We consider simultaneous Waring decompositions: Given forms $ f_d $ of degrees $ kd $, $ (d = 2,3 )$, which admit a representation as $ d $-th power sums of $ k $-forms $ q_1,\ldots,q_m $, when is it possible to reconstruct the addends $ q_1,\ldots,q_m $ from the power sums $ f_d $? Such powers-of-forms decompositions model the moment problem for mixtures of centered Gaussians. The novel approach of this paper is to use semidefinite programming in order to perform a reduction to tensor decomposition. The proposed method works on typical parameter sets at least as long as $ m\leq n-1 $, where $ m $ is the rank of the decomposition and $ n $ is the number of variables. While provably not tight, this analysis still gives the currently best known rank threshold for decomposing third order powers-of-forms, improving on previous work in both asymptotics and constant factors. Our algorithm can produce proofs of uniqueness for specific decompositions. A numerical study is conducted on Gaussian random trace-free quadratics, giving evidence that the success probability converges to $ 1 $ in an average case setting, as long as $ m = n $ and $ n\to \infty $. Some evidence is given that the algorithm also succeeds on instances of rank $ m = \Theta(n^2) $.