发表机构
Chongqing University of Technology（重庆理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出独立伯努利采样方案，证明次采样矩阵的前导奇异向量逼近低秩对称矩阵目标子空间，并给出次采样Davis-Kahan界，揭示计算成本与统计误差的权衡，实现大规模谱分析。
AI 中文摘要
Davis-Kahan定理是谱分析中的一个基本工具，它提供了对称矩阵及其扰动之间特征空间距离的定量控制。然而，当矩阵维度很大时，计算前导特征向量在计算上代价高昂，限制了谱方法在现代大规模应用中的实际使用。本文通过提出一种独立的伯努利采样方案来解决这一问题，并证明了次采样矩阵的前导左奇异向量能够忠实地逼近低秩对称矩阵的目标子空间。我们的主要结果是一个次采样Davis-Kahan界，它给出了直接依赖于采样概率的显式误差界。该界揭示了权衡关系：计算成本随采样概率线性增长，而统计误差随采样概率的平方根倒数变化。因此，我们的结果将Davis-Kahan定理推广到次采样设置，使得大规模对称矩阵的可扩展谱分析成为可能。
英文摘要
The Davis-Kahan theorem is a fundamental tool in spectral analysis, providing quantitative control over the distance between the eigenspaces of a symmetric matrix and its perturbation. However, when the matrix dimension is large, computing leading eigenvectors is computationally expensive, limiting the practical use of spectral methods in modern large-scale applications. This paper addresses this problem by proposing an independent Bernoulli sampling scheme and proves that the leading left singular vectors of the subsampled matrix faithfully approximate the target subspace of a low-rank symmetric matrix. Our main result is a subsampled Davis-Kahan bound that gives an explicit error bound depending directly on the sampling probability. The bound reveals the trade-off: the computational cost scales linearly with the sampling probability, while the statistical error scales as the inverse square root of the sampling probability. Our result thus extends the Davis-Kahan theorem to the subsampled setting, enabling scalable spectral analysis of large-scale symmetric matrices.
Comments24 pages
