Abstract
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.
Highlights
In the context of algebraic statistics [19], moments of probability distributions have recently been explored from an algebraic and geometric point of view [1,4,11,13]
One of the main applications for statistical inference is in the context of the method of moments, which matches the distribution’s moments to moment estimates obtained from a sample
Algebraic identifiability of SeckH (Gn,d ) means that a general homoscedastic Gaussian mixture in the homoscedastic k-secant variety is identifiable from its moments up to order d in the sense that only finitely many Gaussian mixture distributions share the same moments up to order d, whereas we reserve the term rationally identifiable if a general fiber consists of a single point, up to label swapping
Summary
In the context of algebraic statistics [19], moments of probability distributions have recently been explored from an algebraic and geometric point of view [1,4,11,13]. The moments, up to order d, of homoscedastic Gaussian mixtures are still polynomials in the parameters (the means and the covariance matrix), and form the moment variety SeckH (Gn,d ). This is a set of special k-secants inside the secant variety Seck (Gn,d ). As is often observed [1,4,13], a change of coordinates to cumulants tends to yield simpler representations and faster computations This is the case here and we study the cumulant varieties of the homoscedastic Gaussian mixtures. We conclude with a summary of results and list further research directions
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