🤖 AI Summary
This study addresses the detectability of latent phase-coherent structures in observations on the unit circle, distinguishing between uniform and non-uniform distributions with planted signals. A hypothesis testing framework is developed for circular data, employing hard clustering of arcs under a “flat” null model and von Mises mixture models under a “community” alternative. The work establishes nearly tight information-theoretic detection thresholds for four distinct planted signal models, accommodating unknown location parameters and varying correlation structures. By integrating hypothesis testing, information-theoretic lower bounds, concentration inequalities, and circular statistical modeling, the authors derive matching necessary and sufficient conditions for detection—up to constant factors (and logarithmic terms in some cases)—thereby fully characterizing the detection phase transition boundary across all models.
📝 Abstract
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.