  In this paper, we consider the problem of partitioning a small data sample
drawn from a mixture of $k$ product distributions. We are interested in the
case that individual features are of low average quality $\gamma$, and we want
to use as few of them as possible to correctly partition the sample. We analyze
a spectral technique that is able to approximately optimize the total data
size--the product of number of data points $n$ and the number of features
$K$--needed to correctly perform this partitioning as a function of $1/\gamma$
for $K>n$. Our goal is motivated by an application in clustering individuals
according to their population of origin using markers, when the divergence
between any two of the populations is small.
