杠杆分数的条件与QR分解计算
Conditioning of Leverage Scores and Computation by QR Decomposition
AI总结：
研究分析了杠杆分数在矩阵扰动下的相对变化，通过QR分解的扰动结果推导出三种边界条件，揭示了不同扰动类型对杠杆分数精度的影响。
摘要：
全列秩矩阵A的杠杆分数是其范围(A)的任一正交基的行范数平方。我们证明，若A和A+ΔA的主角都小，且其幅度大，则对应的杠杆分数在相对意义上接近。我们还展示了三种基于QR分解扰动结果的边界条件，表明个体杠杆分数的相对差异强烈依赖于ΔA的扰动类型。这些边界条件表明，若ΔA是一般扰动，则个体杠杆分数的相对精度取决于其幅度和A的二范数条件数；若ΔA具有与A相同的范数行缩放，则取决于A的二范数条件数；若ΔA是逐元素行缩放扰动，则（近似）不依赖条件数或杠杆分数幅度。数值实验验证了这些边界的准确性和定量精度。
英文摘要：
The leverage scores of a full-column rank matrix A are the squared row norms of any orthonormal basis for range(A). We show that corresponding leverage scores of two matrices A and A + ΔA are close in the relative sense, if they have large magnitude and if all principal angles between the column spaces of A and A + ΔA are small. We also show three classes of bounds that are based on perturbation results of QR decompositions. They demonstrate that relative differences between individual leverage scores strongly depend on the particular type of perturbation ΔA. The bounds imply that the relative accuracy of an individual leverage score depends on: its magnitude and the two-norm condition of A, if ΔA is a general perturbation; the two-norm condition number of A, if ΔA is a perturbation with the same norm-wise row-scaling as A; (to first order) neither condition number nor leverage score magnitude, if ΔA is a component-wise row-scaled perturbation. Numerical experiments confirm the qualitative and quantitative accuracy of our bounds.
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