Sharper dimension-free bounds on the Frobenius distance between sample covariance and its expectation

Understand

We study properties of a sample covariance estimate $\widehat \Sigma$ given a finite sample of $n$ i.i.d.

• centered random elements in $\R^d$ with the covariance matrix $\Sigma$.
• We derive dimension-free bounds on the squared Frobenius norm of $(\widehat\Sigma - \Sigma)$ under reasonable assumptions.
• For instance, we show that $\smash{\|\widehat\Sigma - \Sigma\|_{\rm F}^2}$ differs from its expectation by at most $\smash{\mathcal O({\rm{Tr}}(\Sigma^2) / n)}$ with overwhelming probability, which is a significant improvement over the existing results.