  We consider the problem of estimating the discrete clustering structures
under the Sub-Gaussian Mixture Model. Our main results establish a hidden
integrality property of a semidefinite programming (SDP) relaxation for this
problem: while the optimal solution to the SDP is not integer-valued in
general, its estimation error can be upper bounded by that of an idealized
integer program. The error of the integer program, and hence that of the SDP,
are further shown to decay exponentially in the signal-to-noise ratio. In
addition, we show that the SDP relaxation is robust under the semi-random
setting in which an adversary can modify the data generated from the mixture
model. In particular, we generalize the hidden integrality property to the
semi-random model and thereby show that SDP achieves the optimal error bound in
this setting. These results together highlight the "global-to-local" mechanism
that drives the performance of the SDP relaxation.
  To the best of our knowledge, our result is the first exponentially decaying
error bound for convex relaxations of mixture models. A corollary of our
results shows that in certain regimes the SDP solutions are in fact integral
and exact. More generally, our results establish sufficient conditions for the
SDP to correctly recover the cluster memberships of $(1-\delta)$ fraction of
the points for any $\delta\in(0,1)$. As a special case, we show that under the
$d$-dimensional Stochastic Ball Model, SDP achieves non-trivial (sometimes
exact) recovery when the center separation is as small as $\sqrt{1/d}$, which
improves upon previous exact recovery results that require constant separation.
