Spectral Concentration and Recovery in Sparse High-Dimensional Random Geometric Graphs

2026-07-15

AI Summary

This study addresses the recovery of latent geometric structure in sparse high-dimensional random geometric graphs generated via inner-product thresholding, focusing on spectral analysis challenges under spherical and Gaussian settings. To overcome the failure of conventional spectral methods caused by edge dependencies in the adjacency matrix, the authors introduce a novel analytical framework combining orthogonal polynomial expansions, decoupling techniques, and matrix concentration inequalities, thereby circumventing limitations inherent to trace moment methods. Under weaker assumptions, they establish improved spectral norm bounds that substantially lower the dimensional threshold required for geometric recovery; specifically, when $np \gg \log n$, the estimation errors for both latent vectors and the Gram matrix vanish asymptotically. Furthermore, in a Gaussian mixture block model, they achieve exact polynomial-time cluster recovery via semidefinite programming for the first time, under optimal connectivity scaling and moderate separation conditions.

Abstract

We study sparse random geometric graphs generated by connecting pairs of high-dimensional vectors whose inner product exceeds a threshold. The latent vectors are sampled either uniformly from the sphere or from a standard Gaussian distribution. Although every edge appears with probability $p$, the edges are dependent through their shared latent vectors. For the spherical model, at the connectivity scale $np=Ω(\log n)$, we prove $\|A-\mathbb E A\|=O\left(\sqrt{np\log n}+npτ\right)$, with high probability, where $τ$ is the cap threshold. This sharpens the spectral norm bound of Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions. An analogous result holds for the Gaussian model after removing the fluctuations of the vector norms, yielding improved global synchronization guarantees for the homogeneous Kuramoto model. We then recover the latent geometry from the leading eigenspace. When $np\gg\log n$, both the latent vector and relative Gram matrix errors vanish provided $d\ll np\log(1/p)/\log n$. The required lower dimension is only $d\gg\log(1/p)$ for the spherical model and $d\gg\log^2(1/p)\log n$ for the Gaussian model, improving the recovery guarantees of Li and Schramm (2023). Finally, we prove the first exact recovery result for the Gaussian mixture block model of Li and Schramm (2023). At the optimal connectivity scale $np=Ω(\log n)$, a polynomial-time semidefinite program exactly recovers all labels in a moderate-separation regime, whereas larger separation makes exact recovery impossible because isolated vertices appear with high probability. Our proofs combine orthogonal polynomial expansions, decoupling, and matrix concentration, avoiding the trace-moment arguments used in previous work.

Problem

• random geometric graphs
• latent geometry recovery
• spectral concentration
• high-dimensional inference
• exact recovery

Innovation

• spectral concentration
• random geometric graphs
• latent geometry recovery
• matrix concentration
• semidefinite programming