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

arXiv: https://arxiv.org/abs/2607.14304 · HTML: https://mlsanity.com/paper/2607.14304/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 71 references that are in the map.
- First posted on arXiv 15 Jul 2026.

## Builds on (most cited first, 4 of 4)

- [Spectral clustering in the Gaussian mixture block model](https://mlsanity.com/paper/2305.00979/) (2023)
- [A general technique for approximating high-dimensional empirical kernel matrices](https://mlsanity.com/paper/2511.03892/) (2025)
- [Exact Label Recovery in Euclidean Random Graphs](https://mlsanity.com/paper/2407.11163/) (2024)
- [Random geometric graphs with smooth kernels: sharp detection threshold and a spectral conjecture](https://mlsanity.com/paper/2602.14998/) (2026)

## Built on by (most cited first, 1 of 1)

- [Recovery of latent inner products from an anisotropic Gaussian random geometric graph](https://mlsanity.com/paper/2607.23723/) (2026)

## Abstract (arXiv)

We study sparse threshold random geometric graphs generated by high-dimensional spherical or Gaussian latent vectors. Although each edge has marginal probability p, shared latent variables make the adjacency entries dependent. At the connectivity scale np=Omega(log n), the spherical adjacency matrix satisfies, with high probability,|A-mathbb E A|op=Oleft(√nplog n+npτright), where τ is the cap threshold; an analogous estimate holds for Gaussian vectors after controlling radial fluctuations. This sharpens the spectral bound in Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions and strengthens the global-synchronization guarantee of Abdalla, Bandeira, and Invernizzi (2024) for the homogeneous Kuramoto model. The leading eigenspace also estimates the latent geometry. When npgglog n, vector and relative Gram-matrix errors vanish forlog(1/p)ll dll nplog(1/p)/log n in the spherical model and log2(1/p)log nll dll nplog(1/p)/log n in the Gaussian model, improving the recovery conditions of Li and Schramm (2023). For the Gaussian mixture block model introduced there, a polynomial-time semidefinite program gives, to our knowledge, the first exact-recovery guarantee at the connectivity scale in a moderate-separation regime. At much larger separation, fixed edge density creates isolated vertices and makes exact recovery impossible. Our reusable decoupling and matrix concentration framework avoids trace-moment methods and applies broadly to random graph models with latent vectors.