Threshold for Detecting High Dimensional Geometry in Anisotropic Random Geometric Graphs

06/29/2022

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by   Matthew Brennan, et al.

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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 Erdős-Rényi graph with the same edge probability. If n is the number of vertices and α is the vector of eigenvalues, Eldan and Mikulincer show that detection is possible when n^3 ≫ (α_2/α_3)^6 and impossible when n^3 ≪ (α_2/α_4)^4. We show detection is impossible when n^3 ≪ (α_2/α_3)^6, closing this gap and affirmatively resolving the conjecture of Eldan and Mikulincer.