We consider a group synchronization problem with multiple frequencies which involves observing pairwise relative measurements of group elements on multiple frequency channels, corrupted by Gaussian noise. We study the computational phase transition in the problem of detecting whether a structured signal is present in such observations by analyzing low-degree polynomial algorithms. We show that, assuming the low-degree conjecture, in synchronization models over arbitrary finite groups as well as over the circle group $SO(2)$, a simple spectral algorithm is optimal among algorithms of runtime $\exp(\tilde{\Omega}(n^{1/3}))$ for detection from an observation including a constant number of frequencies. Combined with an upper bound for the statistical threshold shown in Perry et al., our results indicate the presence of a statistical-to-computational gap in such models with a sufficiently large number of frequencies.

翻译：我们考虑一个包含多个频率的群同步问题，该问题涉及在多频率通道上观测被高斯噪声污染的群元素成对相对测量值。通过分析低阶多项式算法，我们研究了从这类观测中检测结构化信号是否存在时的计算相变。我们证明，在低度猜想成立的假设下，对于任意有限群及圆群$SO(2)$上的同步模型，当观测包含固定数量的频率时，一种简单的谱算法在运行时间为$\exp(\tilde{\Omega}(n^{1/3}))$的检测算法中达到最优。结合Perry等人给出的统计阈值上界，我们的结果表明，在具有足够多频率的此类模型中存在统计-计算差距。