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 an 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.

从其已知的有噪声的成对差异估计一组未知角度的角度同步问题出现在各种应用中。可以将其重新构造为涉及图拉普拉斯矩阵的图的优化问题。我们考虑此问题的一般加权版本，其中，噪声对的影响在不同的条目对之间有所不同，并且其中的一些差异已被完全消除。该版本例如出现在密码学中。我们研究了解决该问题的两种常用方法，即特征向量松弛和半定凸松弛。尽管两种方法都可以使用某些恢复保证，但它们的性能要么不令人满意，要么仅限于未加权的图表。我们缩小了差距，