Papers › Unique powers-of-forms decompositions from simple Gram spectrahedra
Unique powers-of-forms decompositions from simple Gram spectrahedra
Alexander Taveira Blomenhofer
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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 q₁,…,qₘ, when is it possible to reconstruct the addends q₁,…,qₘ 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≤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→∞. Some evidence is given that the algorithm also succeeds on instances of rank m = Θ(n²).
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