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

翻译：协方差矩阵具有稀疏矩阵加低秩扰动结构的数据模型在数据科学应用中普遍存在。为利用此类结构，算法通常需要避免代价高昂的矩阵计算（此类计算往往需要立方级时间和平方级存储）。这一目标常通过执行保持此类结构的运算来实现，例如利用Sherman-Morrison-Woodbury公式进行矩阵求逆。本文研究矩阵平方根及逆平方根运算。针对矩阵的低秩扰动，我们论证了（逆）平方根存在低秩近似修正，并通过建立真实修正特征值的几何衰减界来证明该存在性。进而将修正问题表述为代数Riccati方程的解，探讨如何计算该方程的低秩解。我们分析了近似求解代数Riccati方程所产生的近似误差，给出了谱范数和Frobenius范数下的前向与后向误差界。最后，我们描述了所提算法的若干应用场景，并通过数值实验验证其有效性。