A Lanczos-Based Algorithmic Approach for Spike Detection in Large Sample Covariance Matrices
Charbel Abi Younes, Xiucai Ding, Thomas Trogdon
TL;DR
This work tackles estimating the number of spikes in high-dimensional spiked covariance models without computing the full spectrum of the sample covariance matrix $W=YY^*$, where $Y=\Sigma^{1/2}X$. The authors develop a Lanczos-based framework that samples random directions on the sphere to form eigenvector spectral distributions (VESDs) and leverages a fixed-point Stieltjes transform expressed via a continued fraction derived from the Jacobi and Cholesky structure; this yields a robust estimator for the spiked spectral distribution and, in particular, the spike count by counting poles beyond the ASD support. They establish consistency and concentration results for the estimators under a random-matrix local-law regime, showing $O(N^{-1/2})$-level accuracy for edge estimates and pole locations with $n=O(\\log N)$ Lanczos steps, and demonstrate computational efficiency on large-scale problems. Numerically, the method achieves accurate ASD density estimation and spike detection with substantial speedups over eigenvalue-based approaches, and it remains robust to various population covariances, making it attractive for large-scale high-dimensional inference. The work thus provides a scalable, theory-backed alternative for spike detection in modern data applications where eigen-decomposition is prohibitive.
Abstract
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.