跨变量矩阵表单值的最佳清洁 (Optimal cleaning for singular values of cross-covariance matrices)

from arxiv, 36 pages, 6 figures. In v2: details added in some proofs and remark about estimator convergence added in the Optimality section (Sect. 1.3). In v3: added details about the global effect of error in estimating ideally cleaned singular values. In v4: typos corrected and comments added. In v5: new numerical results where added, as well as a short discussion on computational complexity

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 $T/n$ and $T/p$ 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.

翻译：我们给出了一个新的算法来估算交叉变量矩阵 $\ mathbb{E} XY $($X\ in\ mathbb{R}}$($Y)$($Y_in\mathb{R ⁇ p$($Y_in\mathb{R ⁇ p$)) 的两大维信号 $( mathbb{E} XY $) 的 XY$($X\in\in\ mathbb{R}}$($Y$ $) 。当一对一对一对一(X,Y) 的观测数量很大, 但用美元/n美元和美元/ p$/ p$($) 不应该是小的时, 在无药制度下, $n, p, t$($, p, T$) 很大, 概率很高, 这个算法是Frobenius 规范的最佳方法, 用于在旋转不定的估量者之间, 即通过清除单值而使单矢量不变的矢中从经验估计得出的估计者。