We present a comprehensive analysis of singular vector and singular subspace perturbations in the signal-plus-noise matrix model with random Gaussian noise. 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, building on previous results by O'Rourke, Vu, and the author. Our analysis provides fine-grained insights, including $\ell_\infty$ bounds for singular vectors, $\ell_{2, \infty}$ bounds for singular subspaces, and results for linear and bilinear functions of singular vectors. Additionally, we derive $\ell_{2,\infty}$ bounds on perturbed singular vectors, taking into account the weighting by their corresponding singular values. Finally, we explore practical implications of these results in the Gaussian mixture model and the submatrix localization problem.

翻译：本文对含随机高斯噪声的信号加噪声矩阵模型中的奇异向量及奇异子空间扰动进行了全面分析。在假设信号矩阵为低秩的前提下，我们基于O'Rourke、Vu及作者先前的研究成果，以完全广义化的方式扩展了Davis-Kahan-Wedin定理，使其适用于任何酉不变矩阵范数。我们的分析提供了细粒度的理论结果，包括奇异向量的$\ell_\infty$范数界、奇异子空间的$\ell_{2, \infty}$范数界，以及奇异向量线性与双线性函数的相关结论。此外，通过考虑对应奇异值的加权效应，我们推导了扰动奇异向量的$\ell_{2,\infty}$范数界。最后，我们探讨了这些结果在高斯混合模型及子矩阵定位问题中的实际应用价值。