# Universality for eigenvalue algorithms on sample covariance matrices

## TL;DR

This paper establishes a universal statistical behavior for the iteration count of eigenvalue algorithms applied to random sample covariance matrices, providing complexity estimates that hold with high probability.

## Contribution

It proves a universal limit theorem for the halting time of key eigenvalue algorithms on sample covariance matrices, linking algorithm complexity to random matrix theory results.

## Key findings

- Universal limit theorem for halting time of eigenvalue algorithms
- High-probability complexity estimates for random covariance matrices
- Application of eigenvalue and eigenvector statistics to algorithm analysis

## Abstract

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).