  Motivated by applications to group synchronization and quadratic assignment
on random data, we study a general problem of Bayesian inference of an unknown
``signal'' belonging to a high-dimensional compact group, given noisy pairwise
observations of a featurization of this signal. We establish a quantitative
comparison between the signal-observation mutual information in any such
problem with that in a simpler model with linear observations, using
interpolation methods. For group synchronization, our result proves a replica
formula for the asymptotic mutual information and Bayes-optimal
mean-squared-error. Via analyses of this replica formula, we show that the
conjectural phase transition threshold for computationally-efficient weak
recovery of the signal is determined by a classification of the
real-irreducible components of the observed group representation(s), and we
fully characterize the information-theoretic limits of estimation in the
example of angular/phase synchronization over $SO(2)$/$U(1)$. For quadratic
assignment, we study observations given by a kernel matrix of pairwise
similarities and a randomly permutated and noisy counterpart, and we show in a
bounded signal-to-noise regime that the asymptotic mutual information coincides
with that in a Bayesian spiked model with i.i.d. signal prior.
