标题： 噪声下的低秩逆近似扰动界 显示英文标题

标题： Perturbation Bounds for Low-Rank Inverse Approximations under Noise

摘要： 低秩伪逆在可扩展机器学习、优化和科学计算中被广泛用于逼近矩阵逆。 然而，现实世界的矩阵通常会受到噪声的影响，这些噪声来源于采样、草图和量化。 低秩逆逼近的谱范数鲁棒性仍然理解不足。 我们系统地研究了对于一个$n\times n$对称矩阵$A$的谱范数误差$\| (\tilde{A}^{-1})_p - A_p^{-1} \|$，其中$A_p^{-1}$表示$A^{-1}$的最佳秩\(p\)逼近，而$\tilde{A} = A + E$是一个噪声观测值。 在对噪声的温和假设下，我们推导出精确的非渐近扰动界，揭示了误差如何随着特征值间隙、谱衰减以及噪声与$A$低曲率方向的对齐程度而变化。 我们的分析引入了对\emph{非整函数}函数$f(z) = 1/z$的一种新颖的积分技术应用，得出的界限比经典全逆界的简单适应方法提高了多达$\sqrt{n}$倍。 实证上，我们的界限在各种真实和合成矩阵中紧密跟踪真实的扰动误差，而基于经典结果的估计则往往显著高估。 这些发现为嘈杂计算环境中的低秩逆近似提供了实用且谱感知的保证。 显示英文摘要

摘要： Low-rank pseudoinverses are widely used to approximate matrix inverses in scalable machine learning, optimization, and scientific computing. However, real-world matrices are often observed with noise, arising from sampling, sketching, and quantization. The spectral-norm robustness of low-rank inverse approximations remains poorly understood. We systematically study the spectral-norm error $\| (\tilde{A}^{-1})_p - A_p^{-1} \|$ for an $n\times n$ symmetric matrix $A$, where $A_p^{-1}$ denotes the best rank-\(p\) approximation of $A^{-1}$, and $\tilde{A} = A + E$ is a noisy observation. Under mild assumptions on the noise, we derive sharp non-asymptotic perturbation bounds that reveal how the error scales with the eigengap, spectral decay, and noise alignment with low-curvature directions of $A$. Our analysis introduces a novel application of contour integral techniques to the \emph{non-entire} function $f(z) = 1/z$, yielding bounds that improve over naive adaptations of classical full-inverse bounds by up to a factor of $\sqrt{n}$. Empirically, our bounds closely track the true perturbation error across a variety of real-world and synthetic matrices, while estimates based on classical results tend to significantly overpredict. These findings offer practical, spectrum-aware guarantees for low-rank inverse approximations in noisy computational environments.