  The synchronization problem over the special orthogonal group $SO(d)$
consists of estimating a set of unknown rotations $R_1,R_2,...,R_n$ from noisy
measurements of a subset of their pairwise ratios $R_{i}^{-1}R_{j}$. The
problem has found applications in computer vision, computer graphics, and
sensor network localization, among others. Its least squares solution can be
approximated by either spectral relaxation or semidefinite programming followed
by a rounding procedure, analogous to the approximation algorithms of
\textsc{Max-Cut}. The contribution of this paper is three-fold: First, we
introduce a robust penalty function involving the sum of unsquared deviations
and derive a relaxation that leads to a convex optimization problem; Second, we
apply the alternating direction method to minimize the penalty function;
Finally, under a specific model of the measurement noise and for both complete
and random measurement graphs, we prove that the rotations are exactly and
stably recovered, exhibiting a phase transition behavior in terms of the
proportion of noisy measurements. Numerical simulations confirm the phase
transition behavior for our method as well as its improved accuracy compared to
existing methods.
