The Noise-sensitivity phase transition in spectral group synchronization over compact groups

In Group Synchronization, one attempts to recover a collection of unknown group elements from noisy measurements of their pairwise differences. Several important problems in vision and data analysis reduce to group synchronization over various compact groups. Spectral Group Synchronization is a commonly used, robust algorithm for solving group synchronization problems, which relies on diagonalization of a block matrix whose blocks are matrix representations of the measured pairwise differences. Assuming uniformly distributed measurement errors, we present a rigorous analysis of the accuracy and noise sensitivity of spectral group synchronization algorithms over any compact group. We identify a noise threshold above which the performance of the algorithm completely breaks down. Below the threshold, we calculate an asymptotically exact formula for the accuracy, up to the rounding error, as a function of the noise level. We also provide a consistent risk estimate, allowing practitioners to estimate the method's accuracy from available measurements.