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.

翻译：受群同步和随机数据上的二次分配问题应用的启发，我们研究了一个通用贝叶斯推断问题：给定高维紧致群中未知"信号"的特征化成对噪声观测，推断该信号所属的群。通过插值方法，我们建立了此类问题中信号-观测互信息与线性观测简化模型之间的定量比较。对于群同步问题，我们的结果证明了渐近互信息和贝叶斯最优均方误差的副本公式。通过分析该副本公式，我们表明信号弱恢复的计算有效猜想相变阈值由观测群表示的真实不可约分量分类决定，并完整刻画了$SO(2)$/$U(1)$角/相位同步估计的信息论极限。对于二次分配问题，我们研究了由成对相似性核矩阵及其随机置换噪声对应矩阵组成的观测，并在有界信噪比条件下证明渐近互信息与具有独立同分布信号先验的贝叶斯尖峰模型一致。