Threshold for Detecting High Dimensional Geometry in Anisotropic Random Geometric Graphs

TL;DR

This paper establishes the precise threshold for detecting high-dimensional anisotropic geometric structures in random graphs, resolving a conjecture and clarifying when such geometric signals can be distinguished from Erdős-Rényi graphs.

Contribution

It proves the detection impossibility in the previously unresolved regime, completing the theoretical understanding of geometric detection thresholds.

Findings

Detection is impossible when n^3 << (||α||_2/||α||_3)^6

Closes the gap in the detection threshold analysis

Confirms the conjecture of Eldan and Mikulincer

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, 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.