Hypothesis testing problems for circular data are formulated, where observations take values on the unit circle and may contain a hidden, phase-coherent structure. Under the null, the data are independent uniform on the unit circle; under the alternative, either (i) a planted subset of size K concentrates around an unknown phase (the flat setting), or (ii) a planted community of size k induces coherence among the edges of a complete graph (the community setting). In each of the two settings, two circular signal distributions are considered: a hard-cluster distribution, where correlated planted observations lie in an arc of known length and unknown location, and a von Mises distribution, where correlated planted observations follow a von Mises distribution with a common unknown location parameter. For each of the four resulting models, nearly matching necessary and sufficient conditions are derived (up to constants and occasional logarithmic factors) for detectability, thereby establishing information-theoretic phase transitions.

翻译：本文提出了针对圆形数据的假设检验问题，其中观测值取单位圆上的值，并可能包含隐藏的相位相干结构。在原假设下，数据为单位圆上的独立均匀分布；在备择假设下，要么（i）一个大小为K的植入子集围绕未知相位集中（平坦设置），要么（ii）一个大小为k的植入社区在完全图的边之间诱导相干性（社区设置）。在这两种设置中，分别考虑了两种圆形信号分布：硬聚类分布（其中相关的植入观测位于已知长度、未知位置的圆弧上）和冯·米塞斯分布（其中相关的植入观测遵循具有共同未知位置参数的冯·米塞斯分布）。针对由此产生的四种模型，我们推导了检测能力所需近乎匹配的必要和充分条件（在常数和偶尔的对数因子范围内），从而建立了信息论相变。