The singular values and vectors of low rank perturbations of large rectangular random matrices
The singular values and vectors of low rank perturbations of large rectangular random matrices
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DOI：
10.1016/j.jmva.2012.04.019
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发表时间：
2012-10-01
影响因子：
1.6
通讯作者：
Nadakuditi, Raj Rao
中科院分区：
文献类型：
--
作者：
Benaych-Georges, Florent;Nadakuditi, Raj Rao
本文研究了大型矩形随机矩阵有限低秩扰动的奇异值和奇异向量。特别地，我们证明了扰动矩阵的极端奇异值的几乎必然收敛性和相应奇异向量的适当投影。非通过一个积分变换，线性化了自由概率下的矩形加性卷积，证明了随机极限值与未扰动矩阵的极限奇异值分布的显式关系理论当且仅当扰动矩阵的奇异值大于某个临界阈值时，扰动矩阵的奇异值的渐近位置与原矩阵的奇异值的渐近位置不同，该临界阈值取决于上述积分变换.我们研究了这种奇异值相变对相应的左、右奇异特征向量的影响，并讨论了奇异值在这些奇异特征向量周围的涨落非随机限制。(C)2012 Elsevier Inc. All rights reserved.
In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix.As in the prequel, where we considered the eigenvalues of Hermitian matrices, the non-random limiting value is shown to depend explicitly on the limiting singular value distribution of the unperturbed matrix via an integral transform that linearizes rectangular additive convolution in free probability theory. The asymptotic position of the extreme singular values of the perturbed matrix differs from that of the original matrix if and only if the singular values of the perturbing matrix are above a certain critical threshold which depends on this same aforementioned integral transform.We examine the consequence of this singular value phase transition on the associated left and right singular eigenvectors and discuss the fluctuations of the singular values around these non-random limits. (C) 2012 Elsevier Inc. All rights reserved.