This paper develops a spatially resolved perturbation theory for singular vectors under high-dimensional separable noise and applies it to data-driven matrix recovery. In the asymptotic regime where the matrix dimensions are proportional and significantly larger than the signal rank, we derive exact leading-order variance formulas for the singular vector perturbation projected onto any spatial patch. The variance decomposes into a spatially non-uniform component governed by the local noise covariance and a spatially uniform component governed by the global noise level. These formulas provide the foundation for the \emph{extended optimal shrinkage and wavelet shrinkage} (e$\mathcal{OWS}$) algorithm, which recovers low-rank matrices satisfying a mixed Hölder condition. The pipeline begins with optimal shrinkage of singular values, then constructs coupled multiscale partition trees on the row and column spaces from the denoised estimate, generating a tensor Haar-Walsh wavelet basis. Spatially adaptive wavelet shrinkage is applied using data-driven, coefficient-level thresholds derived from the perturbation theory. We establish convergence rates that strictly improve upon both optimal shrinkage and wavelet shrinkage applied in isolation. Numerical simulations demonstrate reliable matrix recovery and accurate reconstruction of the underlying singular subspaces, including an application to fetal ECG extraction.

翻译：本文针对高维可分离噪声下的奇异向量建立了空间分辨扰动理论，并将其应用于数据驱动的矩阵恢复。在矩阵维度成比例且显著大于信号秩的渐近体系下，我们推导了投影到任意空间块上的奇异向量扰动的精确主阶方差公式。该方差可分解为受局部噪声协方差控制的空间非均匀分量和受全局噪声水平控制的空间均匀分量。这些公式为**扩展最优收缩与小波收缩**（e$\mathcal{OWS}$）算法提供了理论基础，该算法能够恢复满足混合Hölder条件的低秩矩阵。处理流程从奇异值的最优收缩开始，随后基于去噪估计在行空间与列空间上构建耦合的多尺度划分树，从而生成张量Haar-Walsh小波基。利用从扰动理论导出的数据驱动、系数级阈值，实施空间自适应小波收缩。我们建立了严格的收敛速率，该速率相较于单独应用最优收缩或小波收缩均有显著提升。数值模拟验证了算法在矩阵恢复和底层奇异子空间精确重建方面的可靠性，包括在胎儿心电信号提取中的应用实例。