Asymptotic mutual information in quadratic estimation problems over compact groups

Kaylee Yingxi Yang, Timothy Wee, Zhou Fan

Abstract:

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.