# Recovery of latent inner products from an anisotropic Gaussian random geometric graph

arXiv: https://arxiv.org/abs/2607.23723 · HTML: https://mlsanity.com/paper/2607.23723/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 30 references that are in the map.
- First posted on arXiv 26 Jul 2026.

## Builds on (most cited first, 14 of 14)

- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [Generalization error of random features and kernel methods: hypercontractivity and kernel matrix concentration](https://mlsanity.com/paper/2101.10588/) (2021)
- [Testing for high-dimensional geometry in random graphs](https://mlsanity.com/paper/1411.5713/) (2014)
- [Latent Distance Estimation for Random Geometric Graphs](https://mlsanity.com/paper/1909.06841/) (2019)
- [Phase Transitions for Detecting Latent Geometry in Random Graphs](https://mlsanity.com/paper/1910.14167/) (2019)
- [Spectral clustering in the Gaussian mixture block model](https://mlsanity.com/paper/2305.00979/) (2023)
- [Community detection and percolation of information in a geometric setting](https://mlsanity.com/paper/2006.15574/) (2020)
- [A probabilistic view of latent space graphs and phase transitions](https://mlsanity.com/paper/2110.15886/) (2021)
- [A general technique for approximating high-dimensional empirical kernel matrices](https://mlsanity.com/paper/2511.03892/) (2025)
- [Random geometric graphs with smooth kernels: sharp detection threshold and a spectral conjecture](https://mlsanity.com/paper/2602.14998/) (2026)
- [Information and dimensionality of anisotropic random geometric graphs](https://mlsanity.com/paper/1609.02490/) (2016)
- [Random Geometric Graphs on Euclidean Balls](https://mlsanity.com/paper/2010.13734/) (2020)
- [Impossibility of latent inner product recovery via rate distortion](https://mlsanity.com/paper/2407.11932/) (2024)
- [Spectral Concentration and Recovery in Sparse High-Dimensional Random Geometric Graphs](https://mlsanity.com/paper/2607.14304/) (2026)

## Abstract (arXiv)

We study the problem of recovering latent inner products from a random geometric graph with anisotropic Gaussian latent points. More precisely, for an i.i.d. sample x1, dots, xn ~ N(0,Sigma) where Sigma ∈ ℝd × d, an edge (i,j) is present in the graph if and only if langle xi, xj rangle ≥ zeta for a threshold zeta. We assume the threshold zeta to be chosen such that the average edge density of the graph is of constant order. To address the undesired degree fluctuations amplified by the anisotropy of the latent points, we consider the doubly centered adjacency matrix of the graph, and estimate the latent inner products using a rank-d spectral approximation of the doubly centered matrix. The estimator obtains a mean squared error with a rate involving the stable rank of the covariance matrix Sigma. Notably, the rate of estimation matches the state of the art for the isotropic case Sigma = Id, and permits an ill-conditioned covariance matrix with a diverging condition number. The analysis of the spectral method proceeds via the entrywise Hermite expansion of the doubly centered adjacency matrix with respect to the latent inner products. Instead of the standard trace method, it uses a decoupling argument recently introduced by Kaushik, Romberg, and Muthukumar (2025) to control nonlinear error terms.