Azadkia–Chatterjee’s correlation coefficient adapts to manifold data

In their seminal work, Azadkia and Chatterjee (Ann. Statist. 49 (2021) 3070–3102) initiated graph-based methods for measuring variable dependence strength. By appealing to nearest neighbor graphs based on the Euclidean metric, they gave an elegant solution to a problem of Rényi (Acta Math. Acad. Sci. Hung. 10 (1959) 441–451). This idea was later developed in Deb, Ghosal and Sen (2020) (https://arxiv.org/abs/2010.01768) and the authors there proved that, quite interestingly, Azadkia and Chatterjee’s correlation coefficient can automatically adapt to the manifold structure of the data. This paper furthers their study in terms of calculating the statistic’s limiting variance under independence—showing that it only depends on the manifold dimension—and extending this distribution-free property to a class of metrics beyond the Euclidean.