Angular Synchronization by Eigenvectors and Semidefinite Programming: Analysis and Application to Class Averaging in Cryo-electron Microscopy

The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles θ1, . . . , θn from m noisy measurements of their offsets θi − θj 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 = ` 400 2 ́ offset measurements of which 90% are outliers in less than a second on a commercial laptop. We use random matrix theory to prove that the eigenvector method gives meaningful results whenever the proportion of good offset measurements is greater than q n 2m . We show that the eigenvector method is asymptotically nearly optimal in the sense that it achieves the information theoretic Shannon bound up to a multiplicative factor that depends on the discretization error of the measurements 2π/L, but not on m and n. The angular synchronization problem is related to the combinatorial optimization problem Max2-Lin mod L for maximizing the number of satisfied linear equations mod L with exactly 2 variables in each equation. There already exist known polynomial-time semidefinite programming (SDP) approximation algorithms to Max-2-Lin mod L, but in practice we find such algorithms to be limited to relatively small size problems. We also present other SDP relaxations for angle recovery, drawing similarities with the Goemans-Williamson algorithm for finding the maximum cut in a weighted graph. Our experiments show that the angle recovery by the eigenvector method and by the different SDP relaxations are comparable in their quality, making the eigenvector method preferable due to its much faster running time. We formulate and analyze the problem of finding class averages for the three-dimensional structure determination of macromolecules from cryo-electron microscopy as a particular angular synchronization problem. The angular synchronization problem in class averaging is special in the sense that the underlying graph of offset measurements is a “small-world” graph on the real projective plane RP , a case we analyze in detail. In the angular synchronization problem, the angles can be viewed as elements of the rotation group SO(2) and the offsets as relations among the group elements. We discuss 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.