# How close are the eigenvectors and eigenvalues of the sample and actual covariance matrices?

## TL;DR

This paper establishes non-asymptotic bounds on how closely the eigenvectors and eigenvalues of sample covariance matrices approximate those of the true covariance matrix, depending on the number of samples and distribution properties.

## Contribution

It provides new non-asymptotic concentration bounds for eigenvectors, eigenspaces, and eigenvalues for a broad class of distributions, including finite second moment and bounded support.

## Key findings

- Inner product between sample and true eigenvectors decreases with eigenvalue distance
- Conditions identified for distinguishing principal components with limited samples
- Bounds applicable to distributions with finite second moments and bounded support

## Abstract

How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite second moment and distributions supported in a centered Euclidean ball, we prove that the inner product between eigenvectors of the sample and actual covariance matrices decreases proportionally to the respective eigenvalue distance. Our findings imply non-asymptotic concentration bounds for eigenvectors, eigenspaces, and eigenvalues. They also provide conditions for distinguishing principal components based on a constant number of samples.