  Maximum likelihood estimation problems are, in general, intractable
optimization problems. As a result, it is common to approximate the maximum
likelihood estimator (MLE) using convex relaxations. In some cases, the
relaxation is tight: it recovers the true MLE. Most tightness proofs only apply
to situations where the MLE exactly recovers a planted solution (known to the
analyst). It is then sufficient to establish that the optimality conditions
hold at the planted signal. In this paper, we study an estimation problem
(angular synchronization) for which the MLE is not a simple function of the
planted solution, yet for which the convex relaxation is tight. To establish
tightness in this context, the proof is less direct because the point at which
to verify optimality conditions is not known explicitly.
  Angular synchronization consists in estimating a collection of $n$ phases,
given noisy measurements of the pairwise relative phases. The MLE for angular
synchronization is the solution of a (hard) non-bipartite Grothendieck problem
over the complex numbers. We consider a stochastic model for the data: a
planted signal (that is, a ground truth set of phases) is corrupted with
non-adversarial random noise. Even though the MLE does not coincide with the
planted signal, we show that the classical semidefinite relaxation for it is
tight, with high probability. This holds even for high levels of noise.
