Computing the Square Root of a Low-Rank Perturbation of the Scaled Identity Matrix
Computing the Square Root of a Low-Rank Perturbation of the Scaled Identity Matrix
复制标题
计算缩放单位矩阵的低阶扰动的平方根
DOI：
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发表时间：
2023
影响因子：
1.5
通讯作者：
Xiaobo Liu
中科院分区：
文献类型：
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作者：
M. Fasi;N. Higham;Xiaobo Liu
。机器学习的视觉和优化方法。为了平方根，因为当p = 2时，总和我们还得出了一个新的牛顿迭代，用于计算利用低级结构的平方根。在应用中产生的数值实验表明，新方法比现有替代方案要小得多秩。
. We consider the problem of computing the square root of a perturbation of the scaled identity matrix, A = αI n + UV ∗ , where U and V are n × k matrices with k ≤ n . This problem arises in various applications, including computer vision and optimization methods for machine learning. We derive a new formula for the p th root of A that involves a weighted sum of powers of the p th root of the k × k matrix αI k + V ∗ U . This formula is particularly attractive for the square root, since the sum has just one term when p = 2. We also derive a new class of Newton iterations for computing the square root that exploit the low-rank structure. We test these new methods on random matrices and on positive definite matrices arising in applications. Numerical experiments show that the new approaches can yield much smaller residual than existing alternatives and can be significantly faster when the perturbation UV ∗ has low rank.