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In this paper we present a method for learning the parameters of a mixture of$k$ identical spherical Gaussians in $n$-dimensional space with an arbitrarilysmall separation between the components. Our algorithm is polynomial in allparameters other than $k$. The algorithm is based on an appropriate grid searchover the space of parameters. The theoretical analysis of the algorithm hingeson a reduction of the problem to 1 dimension and showing that two 1-dimensionalmixtures whose densities are close in the $L^2$ norm must have similar meansand mixing coefficients. To produce such a lower bound for the $L^2$ norm interms of the distances between the corresponding means, we analyze the behaviorof the Fourier transform of a mixture of Gaussians in 1 dimension around theorigin, which turns out to be closely related to the properties of theVandermonde matrix obtained from the component means. Analysis of this matrixtogether with basic function approximation results allows us to provide a lowerbound for the norm of the mixture in the Fourier domain.  In recent years much research has been aimed at understanding thecomputational aspects of learning parameters of Gaussians mixture distributionsin high dimension. To the best of our knowledge all existing work on learningparameters of Gaussian mixtures assumes minimum separation between componentsof the mixture which is an increasing function of either the dimension of thespace $n$ or the number of components $k$. In our paper we prove the firstresult showing that parameters of a $n$-dimensional Gaussian mixture model witharbitrarily small component separation can be learned in time polynomial in$n$.

Learning Gaussian Mixtures with Arbitrary Separation