Azadkia-Chatterjee's correlation coefficient adapts to manifold data

Abstract: In their seminal work, Azadkia and Chatterjee (2021) initiated graph-based methods for measuring variable dependence strength. By appealing to nearest neighbor graphs, they gave an elegant solution to a problem of R\'enyi (R\'enyi, 1959). Their idea was later developed in Deb et al. (2020) 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 and showing that it only depends on the manifold dimension.