基于列采样的次线性时间特征向量近似算法
Sublinear Time Eigenvector Approximation via Column Sampling
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中文总结 AI 辅助
该研究提出基于列采样的次线性时间特征向量近似算法，可高效近似大矩阵离群特征向量，成果可应用于量子启发算法框架，还给出基于截断Nyström方法的替代方案。
中文摘要 AI 辅助
我们研究用于近似大矩阵离群特征向量的次线性时间采样方法。主要成果为：对于元素绝对值不超过1的对称矩阵A∈ℝ^{n×n}，我们的算法仅均匀采样$\tilde{O}(\frac{\text{log}\text{ }n}{\text{ε}^4})$列；对于A的任意满足|λ|≥εn的特征值λ，算法输出近似特征向量v，满足$\text{‖}Av - λv\text{‖}_2 \text{≤} εn$。若仅近似最大幅值特征值的特征向量，算法仅采样$\tilde{O}(\frac{\text{log}\text{ }n}{\text{ε}^2})$列。若具备按A的行、列平方范数比例采样的能力，我们给出的类似结果误差界提升为ε‖A‖_F。对于最大特征向量近似，我们证明该界在对数因子范围内是紧的。我们算法的核心特性是输出特征向量由A的少量列张成，每个元素可在poly(log n, 1/ε)时间内快速计算，这使其可应用于[Tang, STOC 2019]的量子启发算法框架，我们在该框架中首次给出具有加性误差ε‖A‖_F的特征向量近似次线性时间经典算法。最后，我们提出一种基于截断Nyström方法的替代方案，该方案虽无法实现近似特征向量元素的poly(log n, 1/ε)时间计算，但对一般对称矩阵实现了近最优样本复杂度，对半正定矩阵则有更优界。技术上，我们的界基于[Bhattacharjee等人'22]和[Swartworth与Woodruff'25]关于通过随机采样近似对称矩阵离群特征值的近期工作，首次证明这些方法可扩展至特征向量估计问题。
英文摘要
We study sublinear time sampling methods for approximating the outlying eigenvectors of large matrices. Our main result is an algorithm that uniformly samples just $\tilde{O}(\log n/ε^4)$ columns of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries bounded in magnitude by $1$, and, for any eigenvalue $λ$ of $A$ with $|λ| \ge εn$, outputs an approximate eigenvector $v$ satisfying $\|Av - λv\|_2 \le εn$. For approximating just the eigenvector of the largest magnitude eigenvalue, our algorithm samples only $\tilde{O}(\log n/ε^2)$ columns. Given the ability to sample rows and columns of $A$ proportional to their squared norms, we give a similar result with an improved error bound of $ε\|A\|_F$. For top eigenvector approximation, we show our bound is tight up to logarithmic terms. A key feature of our algorithms is that the output eigenvectors are spanned by a small number of $A$'s columns, and individual entries can be computed rapidly, in poly(log n, 1/epsilon) time per entry. This makes them applicable in the quantum-inspired algorithms framework of [Tang, STOC 2019], where we give the first sublinear time classical algorithms for eigenvector approximation with additive error $ε\|A\|_F$. Finally, we present an alternative approach, based on a truncated Nystrom method, that, while not allowing poly(log n, 1/epsilon) time entrywise computation of the approximate eigenvectors, achieves near optimal sample complexity for general symmetric matrices, and improved bounds for positive semidefinite matrices. Technically, our bounds build on recent work on approximating the outlying eigenvalues of symmetric matrices via random sampling in [Bhattacharjee et al. '22] and [Swartworth and Woodruff '25]. We demonstrate for the first time that these approaches extend to the problem of eigenvector estimation.
