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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 $d$-thpowers. 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 forany $ n\in \mathbb N $, this map is generically one-to-one (up to permutationsof $ q_1,\ldots, q_m $ and third roots of unity) in two ranges: $m\le {n\choose2} + 1 $ for $n \leq 16$ and $ m\le {n+5 \choose 6}/{n+1 \choose 2}-{n+1\choose 2}-1$ for $n > 16$, thus proving generic identifiability for mixturesof centered Gaussians from their (exact) moments of degree at most $ 6 $. Thefirst result is obtained by studying the explicit geometry of the tangentialcontact locus of the variety of sums of cubes of quadratic forms at concretepoints, while the second result is accomplished using a link between secantnon-defectivity with identifiability. The latter approach generalizes also tosums of $ d $-th powers of $k$-forms for $d \geq 3$ and $k \geq 2$.

Identifiability for mixtures of centered Gaussians and sums of powers of quadratics