Singer A
Department of Mathematics and PACM, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA,
Appl Comput Harmon Anal. 2011 Jan 30;30(1):20-36. doi: 10.1016/j.acha.2010.02.001.
The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles θ(1), …, θ(n) from m noisy measurements of their offsets θ(i) - θ(j) mod 2π. Of particular interest is angle recovery in the presence of many outlier measurements that are uniformly distributed in [0, 2π) and carry no information on the true offsets. We introduce an efficient recovery algorithm for the unknown angles from the top eigenvector of a specially designed Hermitian matrix. The eigenvector method is extremely stable and succeeds even when the number of outliers is exceedingly large. For example, we successfully estimate n = 400 angles from a full set of m=(4002) offset measurements of which 90% are outliers in less than a second on a commercial laptop. The performance of the method is analyzed using random matrix theory and information theory. We discuss the relation of the synchronization problem to the combinatorial optimization problem Max-2-Lin mod L and present a semidefinite relaxation for angle recovery, drawing similarities with the Goemans-Williamson algorithm for finding the maximum cut in a weighted graph. We present extensions of the eigenvector method to other synchronization problems that involve different group structures and their applications, such as the time synchronization problem in distributed networks and the surface reconstruction problems in computer vision and optics.
角度同步问题是要从一组未知角度θ(1), …, θ(n)的m个关于其偏移量θ(i) - θ(j) mod 2π的噪声测量值中获得准确估计（精确到一个常数相加相位）。特别令人感兴趣的是在存在许多异常测量值的情况下进行角度恢复，这些异常测量值在[0, 2π)上均匀分布且不携带关于真实偏移量的信息。我们从一个特别设计的埃尔米特矩阵的顶部特征向量引入了一种用于未知角度的高效恢复算法。特征向量方法极其稳定，即使异常值的数量极大时也能成功。例如，我们在一台商用笔记本电脑上，在不到一秒的时间内，从m = (4002)个偏移测量值的完整集合中成功估计出n = 400个角度，其中90%是异常值。使用随机矩阵理论和信息理论对该方法的性能进行了分析。我们讨论了同步问题与组合优化问题Max - 2 - Lin mod L的关系，并提出了一种用于角度恢复的半定松弛方法，与用于在加权图中找到最大割的戈曼斯 - 威廉姆森算法有相似之处。我们展示了特征向量方法到其他涉及不同群结构及其应用的同步问题的扩展，例如分布式网络中的时间同步问题以及计算机视觉和光学中的表面重建问题。
