  Suppose that we are given independent, identically distributed samples $x_l$
from a mixture $\mu$ of no more than $k$ of $d$-dimensional spherical gaussian
distributions $\mu_i$ with variance $1$, such that the minimum $\ell_2$
distance between two distinct centers $y_l$ and $y_j$ is greater than $\sqrt{d}
\Delta$ for some $c \leq \Delta $, where $c\in (0,1)$ is a small positive
universal constant. We develop a randomized algorithm that learns the centers
$y_l$ of the gaussians, to within an $\ell_2$ distance of $\delta <
\frac{\Delta\sqrt{d}}{2}$ and the weights $w_l$ to within $cw_{min}$ with
probability greater than $1 - \exp(-k/c)$. The number of samples and the
computational time is bounded above by $poly(k, d, \frac{1}{\delta})$. Such a
bound on the sample and computational complexity was previously unknown when
$\omega(1) \leq d \leq O(\log k)$. When $d = O(1)$, this follows from work of
Regev and Vijayaraghavan. These authors also show that the sample complexity of
learning a random mixture of gaussians in a ball of radius $\Theta(\sqrt{d})$
in $d$ dimensions, when $d$ is $\Theta( \log k)$ is at least $poly(k,
\frac{1}{\delta})$, showing that our result is tight in this case.
