Ensemble-based estimates of eigenvector error for empirical covariance matrices

TL;DR

This paper develops a scalable ensemble-based method to estimate eigenvector errors in empirical covariance matrices, revealing that errors are highly heterogeneous and scale as 1/nr^2, which has implications for uncertainty quantification.

Contribution

It introduces a new ensemble-based approach to estimate eigenvector errors without full eigenvalue knowledge, applicable to large covariance matrices.

Findings

Eigenvector error distribution scales as 1/nr^2 for large matrices.

Eigenvector errors are highly heterogeneous across eigenvectors.

Numerical experiments support the theoretical error scaling.

Abstract

Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $\{\lambda_i\}$ and eigenvectors $\{{\bf u}_i\}$ of a covariance matrix are central to such endeavors, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $n$ to obtain empirical eigenvalues $\{\tilde{\lambda}_i\}$ and eigenvectors $\{\tilde{{\bf u}}_i\}$, and therefore understanding the error so introduced is of central importance. We analyze eigenvector error $\|{\bf u}_i - \tilde{{\bf u}}_i \|^2$ while leveraging the assumption that the true covariance matrix having size $p$ is drawn from a matrix ensemble with known spectral properties---particularly, we assume the distribution of population eigenvalues weakly converges as $p\to\infty$ to a spectral density $\rho(\lambda)$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $p$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $\lambda$ and approximate the distribution of the expected square error $r= \mathbb{E}\left[\| {\bf u}_i - \tilde{{\bf u}}_i \|^2\right]$ across the matrix ensemble for all ${\bf u}_i$ associated with $\lambda_i=\lambda$. We find, for example, that for sufficiently large matrix size $p$ and sample size $n>p$, the probability density of $r$ scales as $1/nr^2$. This power-law scaling implies that eigenvector error is extremely heterogeneous---even if $r$ is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments.