Title: Distribution of singular values in large sample cross-covariance matrices Show Chinese title

Title: 大样本交叉协方差矩阵中的奇异值分布

Abstract: For two large matrices ${\mathbf X}$ and ${\mathbf Y}$ with Gaussian i.i.d.\ entries and dimensions $T\times N_X$ and $T\times N_Y$, respectively, we derive the probability distribution of the singular values of $\mathbf{X}^T \mathbf{Y}$ in different parameter regimes. This extends the Marchenko-Pastur result for the distribution of eigenvalues of empirical sample covariance matrices to singular values of empirical cross-covariances. Our results will help to establish statistical significance of cross-correlations in many data-science applications. Show Chinese abstract

Abstract: 对于具有高斯独立同分布元素的两个大矩阵${\mathbf X}$和${\mathbf Y}$，其维度分别为$T\times N_X$和$T\times N_Y$，我们推导了在不同参数范围内矩阵$\mathbf{X}^T \mathbf{Y}$的奇异值的概率分布。 这将 Marchenko-Pastur 结果从经验样本协方差矩阵特征值的分布扩展到了经验交叉协方差的奇异值。 我们的结果将有助于在许多数据科学应用中建立交叉相关性的统计显著性。