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样本协方差矩阵特征向量经验谱分布的收敛AI 解读

The eigenvector empirical spectral distribution (VESD) is a useful tool in studying the limiting behavior of eigenvalues and eigenvectors of covariance matrices. In this paper, we study the convergence rate of the VESD of sample covariance matrices to the deformed Marčenko–Pastur (MP) distribution. Consider sample covariance matrices of the form $\Sigma ^{1/2}XX^{*}\Sigma ^{1/2}$, where $X=(x_{ij})$ is an $M\times N$ random matrix whose entries are independent random…

社交网络标签的统计模型AI 解读

We consider a social network from which one observes not only network structure (i.e., nodes and edges) but also a set of labels (or tags, keywords) for each node (or user). These labels are self-created and closely related to the user’s career status, life style, personal interests, and many others. Thus, they are of great interest for online marketing. To model their joint behavior with network structure, a complete data model is developed. The model is based on the…

相关矩阵的谱边缘普适性AI 解读

Let $\widetilde{X}_{M\times N}$ be a rectangular data matrix with independent real-valued entries $[\widetilde{x}_{ij}]$ satisfying $\mathbb{E} \widetilde{x}_{ij}=0$ and $\mathbb{E} \widetilde{x}^{2}_{ij}=\frac{1}{M}$, $N,M\to\infty$. These entries have a subexponential decay at the tails. We will be working in the regime $N/M=d_{N}$, $\lim_{N\to\infty}d_{N}\neq0,1,\infty$. In this paper we prove the edge universality of correlation matrices ${X}^{\dagger}X$, where the…