标题： 基于Lanczos的大型样本协方差矩阵中尖峰检测的算法方法 显示英文标题

标题： A Lanczos-Based Algorithmic Approach for Spike Detection in Large Sample Covariance Matrices

摘要： 我们提出了一种新的方法来估计广义尖峰协方差模型中尖峰的数量，而无需直接计算样本协方差矩阵的特征值。 该方法基于 Lanczos 算法以及相关 Jacobi 矩阵及其 Cholesky 分解的渐近性质。分析的一个关键方面是将特征向量谱分布解释为其渐近对应物的摄动。Jacobi 矩阵的具体指数型渐近性使得可以通过有限连分数有效地逼近渐近谱分布的 Stieltjes 变换。作为结果，我们也获得了渐近分布密度和异常值位置的估计。 我们为所提出的估计器提供了相容性保证，并证明了它们在高维情形下的收敛性。我们还表明，当应用于标准尖峰协方差模型时，我们的方法在计算效率和运行时间上优于现有方法，同时仍然对奇异总体协方差具有鲁棒性。 显示英文摘要

摘要： We introduce a new approach for estimating the number of spikes in a general class of spiked covariance models without directly computing the eigenvalues of the sample covariance matrix. This approach is based on the Lanczos algorithm and the asymptotic properties of the associated Jacobi matrix and its Cholesky factorization. A key aspect of the analysis is interpreting the eigenvector spectral distribution as a perturbation of its asymptotic counterpart. The specific exponential-type asymptotics of the Jacobi matrix enables an efficient approximation of the Stieltjes transform of the asymptotic spectral distribution via a finite continued fraction. As a consequence, we also obtain estimates for the density of the asymptotic distribution and the location of outliers. We provide consistency guarantees for our proposed estimators, proving their convergence in the high-dimensional regime. We demonstrate that, when applied to standard spiked covariance models, our approach outperforms existing methods in computational efficiency and runtime, while still maintaining robustness to exotic population covariances.