Computing the square root of a low-rank perturbation of the scaled identity matrix

Fasi, Massimiliano and Higham, Nicholas J. and Liu, Xiaobo (2022) Computing the square root of a low-rank perturbation of the scaled identity matrix. [MIMS Preprint]

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Abstract

We consider the problem of computing the square root of a perturbation of the scaled identity matrix, A = α Iₙ + UVᴴ, 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 pth root of A that involves a weighted sum of powers of the pth root of the k × k matrix α Iₖ + VᴴU. 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 a much smaller residual than existing alternatives and can be significantly faster when the perturbation UVᴴ has low rank.

Item Type: MIMS Preprint
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Mr Massimiliano Fasi
Date Deposited: 03 Jun 2022 11:21
Last Modified: 03 Jun 2022 11:21
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2856

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