Optimal cleaning for singular values of cross-covariance matrices

We give a new algorithm for the estimation of the cross-covariance matrix $\mathbb{E} XY'$ of two large dimensional signals $X\in\mathbb{R}^n$, $Y\in\mathbb{R}^p$ in the context where the number $T$ of observations of the pair $(X,Y)$ is itself large, but with $n,p$ non negligible with respect to $T$. This algorithm is optimal among rotationally invariant estimators, i.e. estimators derived from the empirical estimator by cleaning the singular values, while letting singular vectors unchanged. We give an interpretation of the singular value cleaning in terms of overfitting ratios.