  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 arbitrarily
small separation between the components. Our algorithm is polynomial in all
parameters other than $k$. The algorithm is based on an appropriate grid search
over the space of parameters. The theoretical analysis of the algorithm hinges
on a reduction of the problem to 1 dimension and showing that two 1-dimensional
mixtures whose densities are close in the $L^2$ norm must have similar means
and mixing coefficients. To produce such a lower bound for the $L^2$ norm in
terms of the distances between the corresponding means, we analyze the behavior
of the Fourier transform of a mixture of Gaussians in 1 dimension around the
origin, which turns out to be closely related to the properties of the
Vandermonde matrix obtained from the component means. Analysis of this matrix
together with basic function approximation results allows us to provide a lower
bound for the norm of the mixture in the Fourier domain.
  In recent years much research has been aimed at understanding the
computational aspects of learning parameters of Gaussians mixture distributions
in high dimension. To the best of our knowledge all existing work on learning
parameters of Gaussian mixtures assumes minimum separation between components
of the mixture which is an increasing function of either the dimension of the
space $n$ or the number of components $k$. In our paper we prove the first
result showing that parameters of a $n$-dimensional Gaussian mixture model with
arbitrarily small component separation can be learned in time polynomial in
$n$.
