论文标题
对集中高斯的混合物的可识别性和四边形的力量总和
Identifiability for mixtures of centered Gaussians and sums of powers of quadratics
论文作者
论文摘要
我们认为多项式地图的逆问题，该地图将$ n $变量中的二次二次形式的组合发送给其$ d $ th powers的总和。该地图捕获了$ m $中心的$ n $ variate高斯的混合物的时刻问题。在第一个非平凡情况下，$ d = 3 $，我们表明，对于任何$ n \ in \ mathbb n $，此地图通常是一对一的（最多为$ q_1，\ ldots，q_m $，q_m $和unity的第三根根源和第三个根源），在两个范围内：$ m \ m \ m \ le for 2} + n $ n $ n $ n $ s $ n \ m. 6}/{N+1 \选择2} - {N+1 \选择2} -1 $对于$ n> 16 $，因此从其（确切的）学位瞬间以$ 6 $的价格证明了集中高斯的混合物的通用可识别性。第一个结果是通过研究混凝土点的各种二次形式的分支总和的切向接触基因座的显式几何形状，而第二个结果是使用secant非缺陷性（可识别性）之间的链接完成的。后一种方法也概括为$ d $ th powers $ k $ - forms的总和，$ d \ geq 3 $和$ k \ geq 2 $。
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\in \mathbb N $, this map is generically one-to-one (up to permutations of $ q_1,\ldots, q_m $ and third roots of unity) in two ranges: $m\le {n\choose 2} + 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 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 $d \geq 3$ and $k \geq 2$.
