Optimal cleaning for singular values of cross-covariance matrices
Florent Benaych-Georges, Jean‐Philippe Bouchaud, Marc Potters · The Annals of Applied Probability · 2023
We give a new algorithm for the estimation of the cross-covariance matrix EXY′ of two large-dimensional signals X∈Rn, Y∈Rp in the context where the number T of observations of the pair (X,Y) is large but n/T and p/T are not supposed to be small. In the asymptotic regime where n, p, T are large, with high probability, this algorithm is optimal for the Frobenius norm among rotationally invariant estimators, that is, estimators derived from the empirical estimator by cleaning the singular values, while letting singular vectors unchanged.