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 Davis-Kahan-Wedin 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中的奇异向量和奇异子空间扰动进行了全面分析。假设信号矩阵为低秩矩阵，我们以完全推广的方式扩展了Davis-Kahan-Wedin定理，使其适用于任意酉不变矩阵范数，扩展了O'Rourke、Vu及作者先前的结果。我们还获得了精细化的结果，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索。此外，我们探讨了这些发现在高斯混合模型和子矩阵定位问题中的实际应用价值。