Computing the Square Root of a Low-Rank Perturbation of the Scaled Identity Matrix

Abstract

Listen

Translate article

We consider the problem of computing the square root of a perturbation of the scaled identity matrix, , where and are matrices with . This problem arises in various applications, including computer vision and optimization methods for machine learning. We derive a new formula for the th root of that involves a weighted sum of powers of the th root of the matrix . This formula is particularly attractive for the square root, since the sum has just one term when . 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 has low rank.