Detecting Planted Structure in Circular Data

TL;DR

This paper investigates hypothesis testing for circular data with hidden structures, establishing near-optimal conditions for detecting planted signals under various models and distributions.

Contribution

It introduces a comprehensive framework for detecting planted structures in circular data, deriving nearly matching conditions for detectability across multiple models.

Findings

Derived necessary and sufficient conditions for detection

Established information-theoretic phase transitions

Analyzed different distributions and planted structures

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.