Perturbation theory is developed to analyze the impact of noise on data and has been an essential part of numerical analysis. Recently, it has played an important role in designing and analyzing matrix algorithms. One of the most useful tools in this subject, the Davis-Kahan sine theorem, provides an $\ell_2$ error bound on the perturbation of the leading singular vectors (and spaces). We focus on the case when the signal matrix has low rank and the perturbation is random, which occurs often in practice. In an earlier paper, O'Rourke, Wang, and the second author showed that in this case, one can obtain an improved theorem. In particular, the noise-to-gap ratio condition in the original setting can be weakened considerably. In the current paper, we develop an infinity norm version of the O'Rourke-Vu-Wang result. The key ideas in the proof are a new bootstrapping argument and the so-called iterative leave-one-out method, which may be of independent interest. Applying the new bounds, we develop new, simple, and quick algorithms for several well-known problems, such as finding hidden partitions and matrix completion. The core of these new algorithms is the fact that one is now able to quickly approximate certain key objects in the infinity norm, which has critical advantages over approximations in the $\ell_2$ norm, Frobenius norm, or spectral norm.

翻译：扰动理论用于分析噪声对数据的影响，历来是数值分析的重要组成部分。近年来，它在矩阵算法的设计与分析中发挥着关键作用。该领域最有效的工具之一——Davis-Kahan正弦定理——提供了主导奇异向量（及子空间）扰动的ℓ₂误差界。本文聚焦于信号矩阵低秩且扰动随机的常见实践场景。在前期工作中，O'Rourke、Wang与第二作者已证明该场景下可获得改进定理，尤其原始设定中的噪声-间隙比条件可大幅放宽。本文进一步建立了O'Rourke-Vu-Wang结果的无穷范数版本。证明的核心思想在于新型自举论证与迭代留一法（可能具有独立研究价值）。应用新界，我们为若干经典问题（如隐划分发现与矩阵补全）开发了简洁高效的新算法。这些新算法的关键在于：如今可快速近似无穷范数下的特定关键对象，这比ℓ₂范数、Frobenius范数或谱范数近似具有显著优势。