Conditioning of Leverage Scores and Computation by QR Decomposition
Conditioning of Leverage Scores and Computation by QR Decomposition
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通过 QR 分解调节杠杆分数和计算
DOI：
10.1137/140988541
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Thomas Wentworth
中科院分区：
文献类型：
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作者：
J. Holodnak;Ilse C. F. Ipsen;Thomas Wentworth
全列秩矩阵a的杠杆分数是范围(a)的任何标准正交基的平方行规范。我们表明，如果两个矩阵A和A + \Delta A具有较大的幅度，并且A和A + \Delta A的列空间之间的所有主角都很小，则对应的杠杆分数在相对意义上是接近的。我们还展示了基于QR分解的扰动结果的三类界。他们证明了个体杠杆分数之间的相对差异在很大程度上取决于特定类型的扰动\Delta A。边界意味着个体杠杆分数的相对准确性取决于：如果\Delta A是一般扰动，则其大小和A的双范数条件；A的双范数条件数，如果\Delta A是与A具有相同范数方向行尺度的扰动；（到一阶）既不是条件数也不是杠杆分数大小，如果\Delta A是一个组件方向的行尺度扰动。数值实验证实了该边界在定性和定量上的准确性。
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 + \Delta A are close in the relative sense, if they have large magnitude and if all principal angles between the column spaces of A and A + \Delta A 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 \Delta A. The bounds imply that the relative accuracy of an individual leverage score depends on: its magnitude and the two-norm condition of A, if \Delta A is a general perturbation; the two-norm condition number of A, if \Delta A is a perturbation with the same norm-wise row-scaling as A; (to first order) neither condition number nor leverage score magnitude, if \Delta A is a component-wise row-scaled perturbation. Numerical experiments confirm the qualitative and quantitative accuracy of our bounds.
影响因子：
2.1
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通讯作者：
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