  In this paper, we consider the problem of partitioning a small data sample of
size $n$ drawn from a mixture of $2$ sub-gaussian distributions. Our work is
motivated by the application of clustering individuals according to their
population of origin using markers, when the divergence between the two
populations is small. 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 consider semidefinite relaxation
of an integer quadratic program which is formulated essentially as finding the
maximum cut on a graph where edge weights in the cut represent dissimilarity
scores between two nodes based on their features. A small simulation result in
Blum, Coja-Oghlan, Frieze and Zhou (2007, 2009) shows that even when the sample
size $n$ is small, by increasing $p$ so that $np= \Omega(1/\gamma^2)$, one can
classify a mixture of two product populations using the spectral method therein
with success rate reaching an ``oracle'' curve. There the ``oracle'' was
computed assuming that distributions were known, where success rate means the
ratio between correctly classified individuals and the sample size $n$. In this
work, we show the theoretical underpinning of this observed concentration of
measure phenomenon in high dimensions, simultaneously for the semidefinite
optimization goal and the spectral method, where the input is based on the gram
matrix computed from centered data. We allow a full range of tradeoffs between
the sample size and the number of features such that the product of these two
is lower bounded by $1/{\gamma^2}$ so long as the number of features $p$ is
lower bounded by $1/\gamma$.
