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

2026-07-26

AI Summary

This work addresses the problem of recovering the latent inner product structure from random geometric graphs generated by anisotropic Gaussian latent variables. To mitigate degree fluctuations induced by anisotropy, the authors propose an estimator based on doubly centered adjacency matrices combined with rank-$d$ spectral approximation. They further introduce element-wise Hermite expansions and a decoupling technique to control nonlinear approximation errors. Remarkably, the proposed method achieves mean squared error rates comparable to those in the isotropic setting—even when the covariance matrix is ill-conditioned or its condition number diverges. The estimator’s performance is governed by the stable rank of the covariance matrix, attaining the optimal estimation accuracy currently known for this anisotropic regime.

Abstract

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 $x_1, \dots, x_n \sim N(0,Σ)$ where $Σ\in \mathbb{R}^{d \times d}$, an edge $(i,j)$ is present in the graph if and only if $\langle x_i, x_j \rangle \ge ζ$ for a threshold $ζ$. We assume the threshold $ζ$ 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 $Σ$. Notably, the rate of estimation matches the state of the art for the isotropic case $Σ= I_d$, 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.

Problem

• latent inner products
• anisotropic Gaussian
• random geometric graph
• covariance matrix
• graph recovery

Innovation

• anisotropic Gaussian
• random geometric graph
• doubly centered adjacency matrix
• spectral estimation
• Hermite expansion