# Information and dimensionality of anisotropic random geometric graphs

arXiv: https://arxiv.org/abs/1609.02490 · HTML: https://mlsanity.com/paper/1609.02490/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Sep 2016.

## Built on by (most cited first, 2 of 2)

- [Impossibility of latent inner product recovery via rate distortion](https://mlsanity.com/paper/2407.11932/) (2024)
- [Recovery of latent inner products from an anisotropic Gaussian random geometric graph](https://mlsanity.com/paper/2607.23723/) (2026)

## Abstract (arXiv)

This paper deals with the problem of detecting non-isotropic high-dimensional geometric structure in random graphs. Namely, we study a model of a random geometric graph in which vertices correspond to points generated randomly and independently from a non-isotropic d-dimensional Gaussian distribution, and two vertices are connected if the distance between them is smaller than some pre-specified threshold. We derive new notions of dimensionality which depend upon the eigenvalues of the covariance of the Gaussian distribution. If α denotes the vector of eigenvalues, and n is the number of vertices, then the quantities left(frac||α||2||α||3right)6/n3 and left(frac||α||2||α||4right)4/n3 determine upper and lower bounds for the possibility of detection. This generalizes a recent result by Bubeck, Ding, R'acz and the first named author from [BDER14] which shows that the quantity d/n3 determines the boundary of detection for isotropic geometry. Our methods involve Fourier analysis and the theory of characteristic functions to investigate the underlying probabilities of the model. The proof of the lower bound uses information theoretic tools, based on the method presented in [BG15].