发表机构
Department of Mathematics, University of Southern California（数学系，南加州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究稀疏高维随机几何图，通过连接内积超阈值的向量对生成。证明了球面和高斯模型的谱范数界，改进恢复保证，还得到高斯混合块模型的精确恢复结果，采用新证明方法结合多种技术。
AI 中文摘要
我们研究通过连接内积超过阈值的高维向量对生成的稀疏随机几何图。潜在向量从球体或标准高斯分布中均匀采样。尽管每条边出现概率为\(p\)，但边通过共享潜在向量相关。对于球面模型，在连通性尺度\(np = \Omega(\log n)\)时，我们证明\(\|A - \mathbb{E}A\| = O(\sqrt{np\log n} + np\tau)\)，概率很高，其中\(\tau\)是帽阈值，此结果在更弱假设下改进了Liu等人(2023)的谱范数界。高斯模型有类似结果，改进了齐次Kuramoto模型的全局同步保证。然后从主特征空间恢复潜在几何。当\(np\gg\log n\)时，在一定条件下潜在向量和相对Gram矩阵误差消失，改进了Li和Schramm(2023)的恢复保证。最后，我们证明了Li和Schramm(2023)的高斯混合块模型的第一个精确恢复结果。在最优连通性尺度\(np = \Omega(\log n)\)时，多项式时间半定规划在中等分离区域能精确恢复所有标签，更大分离则因孤立顶点大概率出现而无法精确恢复。我们的证明结合了正交多项式展开、解耦和矩阵集中，避免了先前工作中使用的迹矩论证。
英文摘要
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 $np=Ω(\log n)$, the spherical adjacency matrix satisfies, with high probability,$\|A-\mathbb E A\|_{\mathrm{op}}=O\left(\sqrt{np\log n}+npτ\right)$, 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 $np\gg\log n$, vector and relative Gram-matrix errors vanish for$\log(1/p)\ll d\ll np\log(1/p)/\log n$ in the spherical model and $\log^2(1/p)\log n\ll d\ll np\log(1/p)/\log n$ 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.
Comments67 pages, 2 figures
