AI 中文总结
研究从各向异性高斯随机几何图恢复潜在内积问题，考虑图的双中心邻接矩阵，用秩\(d\)谱近似估计潜在内积，其均方误差速率涉及协方差矩阵稳定秩，匹配各向同性情况且允许病态协方差矩阵，分析用新解耦论证控制误差项。
AI 中文摘要
我们研究从具有各向异性高斯潜在点的随机几何图中恢复潜在内积的问题。具体而言，对于独立同分布样本\(x_1, \dots, x_n \sim N(0,\Sigma)\)（\(\Sigma \in \mathbb{R}^{d \times d}\)），当且仅当\(\langle x_i, x_j \rangle \ge \zeta\)（\(\zeta\)为阈值）时图中存在边\((i,j)\)。为解决潜在点各向异性放大的不良度波动问题，我们考虑图的双中心邻接矩阵，并用双中心矩阵的秩\(d\)谱近似估计潜在内积。该估计器得到的均方误差速率涉及协方差矩阵\(\Sigma\)的稳定秩。值得注意的是，估计速率与各向同性情况\(\Sigma = I_d\)的现有技术水平匹配，且允许条件数发散的病态协方差矩阵。谱方法的分析通过双中心邻接矩阵关于潜在内积的逐元素埃尔米特展开进行，使用了Kaushik、Romberg和Muthukumar（2025）最近引入的解耦论证来控制非线性误差项。
英文摘要
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.
Comments41 pages
