Threshold for Detecting High Dimensional Geometry in Anisotropic Random Geometric Graphs

Brice Huang, Matthew Stewart Brennan, Guy Bresler

Abstract:

In the anisotropic random geometric graph model, vertices correspond to points drawn from a high-dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold.
We study when it is possible to hypothesis test between such a graph and an Erd\H{o}s-R\'enyi graph with the same edge probability.
If $n$ is the number of vertices and $\alpha$ is the vector of eigenvalues, \cite{eldan2020information} shows that detection is possible when $n^3 \gg (\|\alpha\|_2/\|\alpha\|_3)^6$ and impossible when $n^3 \ll (\|\alpha\|_2/\|\alpha\|_4)^4$.
We show detection is impossible when $n^3 \ll (\|\alpha\|_2/\|\alpha\|_3)^6$, closing this gap and affirmatively resolving the conjecture of \cite{eldan2020information}.