Analysis of singular subspaces under random perturbations
研究了信号加噪声矩阵模型中奇异向量和奇异子空间在随机高斯噪声下的扰动，推广了Davis-Kahan-Wedin定理到任意酉不变矩阵范数，并给出了ℓ∞和ℓ2,∞界，适用于高斯混合模型和子矩阵定位问题。
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 ℓ∞ bounds for singular vectors, ℓ2,∞ bounds for singular subspaces, and results for linear and bilinear functions of singular vectors. Additionally, we derive ℓ2,∞ 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.