Threshold for Detecting High Dimensional Geometry in Anisotropic Random Geometric Graphs
Abstract
In the anisotropic random geometric graph model, vertices correspond to points drawn from a highdimensional 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 ErdősRényi graph with the same edge probability. If $n$ is the number of vertices and $\alpha$ is the vector of eigenvalues, Eldan and Mikulincer show 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 Eldan and Mikulincer.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.14896
 Bibcode:
 2022arXiv220614896B
 Keywords:

 Mathematics  Statistics Theory;
 Computer Science  Information Theory;
 Mathematics  Probability
 EPrint:
 11 pages, comments welcome