We present a comprehensive analysis of singular vector and singular subspace perturbations in the context of the signal plus random Gaussian noise matrix model. Assuming a low-rank signal matrix, we extend the Wedin-Davis-Kahan theorem in a fully generalized manner, applicable to any unitarily invariant matrix norm, extending previous results of O'Rourke, Vu and the author. We also obtain the fine-grained results, which encompass the $\ell_\infty$ analysis of singular vectors, the $\ell_{2, \infty}$ analysis of singular subspaces, as well as the exploration of linear and bilinear functions related to the singular vectors. Moreover, we explore the practical implications of these findings, in the context of the Gaussian mixture model and the submatrix localization problem.

翻译：我们针对信号加随机高斯噪声矩阵模型下的奇异向量和奇异子空间扰动问题，提出了一种全面的分析方法。基于低秩信号矩阵假设，我们以完全广义的方式推广了Wedin-Davis-Kahan定理，使其适用于任意酉不变矩阵范数，扩展了O'Rourke、Vu及作者的前期成果。我们还获得了细粒度结果，包括奇异向量的$\ell_\infty$分析、奇异子空间的$\ell_{2,\infty}$分析，以及与奇异向量相关的线性和双线性函数的探究。此外，我们在高斯混合模型和子矩阵定位问题的背景下探讨了这些发现的实践意义。