The Angular Synchronization problem consists of estimating a set of unknown angles $\theta_i$ from noisy measurements of a subset of the pairwise differences $\theta_i-\theta_j$. We provide and analyze a spectral method that robustly solves this problem using the information of all measurements in a democratic fashion. Furthermore, we relate the consistency of the measurements with the spectrum of an operator, the graph Connection Laplacian through a Cheeger-type inequality. (joint work with A. Singer (Princeton) and D. Spielman (Yale) ).
Finally, we will address a problem that naturally arises in X-ray microscopy -- the phase retrieval problem. We show how the spectral method to solve synchronization plays a major role on providing stability for a particular method to solve this problem. (joint work with: B. Alexeev (Princeton), D. Mixon (Air Force Inst. Tech.), M. Fickus (Air Force Inst. Tech), and Y. Chen (Princeton) ).