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In the anisotropic random geometric graph model, vertices correspond topoints drawn from a high-dimensional Gaussian distribution and two vertices areconnected if their distance is smaller than a specified threshold. We studywhen it is possible to hypothesis test between such a graph and anErd\H{o}s-R\'enyi graph with the same edge probability. If $n$ is the number ofvertices and $\alpha$ is the vector of eigenvalues, Eldan and Mikulincer showthat detection is possible when $n^3 \gg (\|\alpha\|_2/\|\alpha\|_3)^6$ andimpossible when $n^3 \ll (\|\alpha\|_2/\|\alpha\|_4)^4$. We show detection isimpossible when $n^3 \ll (\|\alpha\|_2/\|\alpha\|_3)^6$, closing this gap andaffirmatively resolving the conjecture of Eldan and Mikulincer.

Threshold for Detecting High Dimensional Geometry in Anisotropic Random Geometric Graphs