2022/04/20 by Alexander Taveira Blomenhofer, Alex Casarotti, Blomenhofer, Alexander Taveira +5 · 1 citation
Computer Science · Mathematics · #14Q20 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Point processes and geometric inequalities #Polynomial and algebraic computation #primary: 15A69
paper · pdf · doi:10.48550/arxiv.2204.09356
openalex publication_date 2022/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the inverse problem for the polynomial map which sends an m-tuple of quadratic forms in n variables to the sum of their d-th powers. This map captures the moment problem for mixtures of m centered n-variate Gaussians. In the first non-trivial case d = 3, we show that for any n∈ \mathbb N , this map is generically one-to-one (up to permutations of q1,…, qm and third roots of unity) in two ranges: m≤ n\choose 2 + 1 for n ≤ 16 and m≤ n+5 \choose 6/n+1 \choose 2-n+1 \choose 2-1 for n > 16, thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most 6 . The first result is obtained by studying the explicit geometry of the tangential contact locus of the variety of sums of cubes of quadratic forms at concrete points, while the second result is accomplished using a link between secant non-defectivity with identifiability. The latter approach generalizes also to sums of d -th powers of k-forms for d ≥ 3 and k ≥ 2.