Low‐rank updates of matrix square roots

Abstract Models in which the covariance matrix has structure of a sparse plus low rank perturbation are ubiquitous data science applications. It is often desirable for algorithms to take advantage such structures, avoiding costly computations that require cubic time and quadratic storage. This accomplished by performing operations maintain example, inversion via Sherman–Morrison–Woodbury formula. In this article, we consider square root inverse operations. Given matrix, argue low‐rank approximate correction (inverse) exists. We do so establishing geometric decay bound on true correction's eigenvalues. then proceed frame as solution an algebraic Riccati equation, discuss how equation can be computed. analyze approximation error incurred when approximately solving providing spectral Frobenius norm forward backward bounds. Finally, describe several applications our algorithms, demonstrate their utility numerical experiments.