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

We consider the problem of computing square root a perturbation scaled identity matrix, , where and are matrices with . This arises in various applications, including computer vision optimization methods for machine learning. derive new formula th that involves weighted sum powers matrix is particularly attractive root, since has just one term when also class Newton iterations exploit low-rank structure. test these on random positive definite arising applications. Numerical experiments show approaches can yield much smaller residual than existing alternatives be significantly faster low rank.