We consider the problem of identifying a mixture of Gaussian distributions with the same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special secants to the Gaussian moment varieties, defined by their natural polynomial parametrization in terms of the model parameters. When the order of the moments is at most three, we prove an analogue of the Alexander–Hirschowitz theorem classifying all cases of homoscedastic Gaussian mixtures that produce defective moment varieties. As a consequence, identifiability is determined when the number of mixed distributions is smaller than the dimension of the space. In the two-component setting, we provide a closed form solution for parameter recovery based on moments up to order four, while in the one-dimensional case we interpret the rank estimation problem in terms of secant varieties of rational normal curves.

我们考虑的问题是，通过高斯分布的矩序直至一定阶数来确定高斯分布的混合体。我们的方法是研究通过将特殊割线带入高斯矩量变种而获得的矩量变种，高斯矩量变种是根据模型参数将其自然多项式参数化定义的。当力矩阶数最多为3时，我们证明了Alexander–Hirschowitz定理的类似物，该定理将产生高阶力矩变种的所有同调高斯混合情况分类。结果，当混合分布的数量小于空间的尺寸时，确定可识别性。在两部分设置中，我们提供了一种封闭形式的解决方案，用于根据直到四阶的矩进行参数恢复，