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 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: 13 Jan 2022 21:08
Last Modified: 13 Jan 2022 21:08
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2842

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