Universality for eigenvalue algorithms on sample covariance matrices
For researchers in numerical linear algebra and random matrix theory, this work establishes rigorous complexity bounds for eigenvalue algorithms in a random setting, though the results are theoretical and incremental in nature.
The paper proves a universal limit theorem for the iteration count of power/inverse power methods and the QR eigenvalue algorithm on random sample covariance matrices, providing complexity estimates that hold with high probability.
We prove a universal limit theorem for the halting time, or iteration count, of the power/inverse power methods and the QR eigenvalue algorithm. Specifically, we analyze the required number of iterations to compute extreme eigenvalues of random, positive-definite sample covariance matrices to within a prescribed tolerance. The universality theorem provides a complexity estimate for the algorithms which, in this random setting, holds with high probability. The method of proof relies on recent results on the statistics of the eigenvalues and eigenvectors of random sample covariance matrices (i.e., delocalization, rigidity and edge universality).