  In Group Synchronization, one attempts to find 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, up to the
rounding error. We identify a Baik-Ben Arous-P\'ech\'e type phase transition in
the noise level, beyond which spectral group synchronization necessarily fails.
Below the phase transition, spectral group synchronization succeeds in
recovering the unknown group elements, but its performance deteriorates with
the noise level. We provide asymptotically exact formulas for the accuracy of
spectral group synchronization below the phase transition, up to the rounding
error. We also provide a consistent risk estimate, allowing practitioners to
estimate the method's accuracy from available measurements.
