计算具有正交行的矩阵的强秩揭示分解
Computing Strong Rank-Revealing Factorizations for Matrices with Orthonormal Rows
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中文总结 AI 辅助
研究具有正交行矩阵的强秩揭示分解，采用斯图尔特枢轴策略与GKS列选择算法结合，可实现良好精度界和基条件，还从两方向扩展框架，包括对GKS近似情况分析及提供随机变体，速度更快。
中文摘要 AI 辅助
我们证明，斯图尔特提出的一种基于比绍夫工作的枢轴策略，应用于具有正交行的矩阵时可计算出强秩揭示分解。与经典的戈卢布、克莱马和斯图尔特（GKS）列选择算法结合使用时，它有助于实现与直接对A应用强秩揭示分解一样好的秩-k近似精度界和基条件。然后，我们从两个方向扩展此框架：（1）在仅可获得右奇异向量近似值时对GKS进行分析；（2）为具有正交行的矩阵提供枢轴策略的随机变体，该变体具有相同的理论保证，但返回所需子集的速度比确定性变体快两个数量级。
英文摘要
We show that a pivoting strategy due to Stewart (based on work by Bischof) computes a strong rank-revealing factorization when applied to a matrix with orthonormal rows. When paired with the classical column selection algorithm of Golub, Klema, and Stewart (GKS) it helps achieve rank-$k$ approximation accuracy bounds and basis conditioning as good as those from applying a strong rank-revealing factorization directly to A. We then extend this framework in two directions: (1) providing analysis of GKS when only approximations of right singular vectors are available and (2) providing a randomized variant of the pivoting strategy for matrices with orthonormal rows that achieves the same theoretical guarantees but can return the desired subset two orders of magnitude faster than the deterministic variant.
