  The angular synchronization problem of estimating a set of unknown angles
from their known noisy pairwise differences arises in various applications. It
can be reformulated as a optimization problem on graphs involving the graph
Laplacian matrix. We consider a general, weighted version of this problem,
where the impact of the noise differs between different pairs of entries and
some of the differences are erased completely; this version arises for example
in ptychography. We study two common approaches for solving this problem,
namely eigenvector relaxation and semidefinite convex relaxation. Although some
recovery guarantees are available for both methods, their performance is either
unsatisfying or restricted to the unweighted graphs. We close this gap,
deriving recovery guarantees for the weighted problem that are completely
analogous to the unweighted version.
