Optimal cleaning for singular values of cross-covariance matrices

Abstract

We give a new algorithm for the estimation of the cross-covariance matrix $\mathbb{E}X{Y}^{\prime }$ 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 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, i.e. estimators derived from the empirical estimator by cleaning the singular values, while letting singular vectors unchanged.

38 pages, 8 figures, 2 tables. In v8: numerical simulations section widely extended, intro partly rephrased, section about overfitting extended, typos corrected, link to github repo for python codes added

Cited by in corpus (1)

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