矩阵平方根的低秩更新
Low-Rank Updates of Matrix Square Roots
AI总结：
针对稀疏矩阵加低秩扰动的协方差结构，本文将平方根及逆平方根校正转化为代数 Riccati 方程，提出可计算的低秩近似并给出误差界与实验验证。
摘要：
协方差矩阵具有稀疏矩阵加低秩扰动结构的模型在数据科学应用中无处不在。算法通常需要利用这种结构，避免往往需要三次时间和二次存储的昂贵矩阵计算。这通常通过执行保持此类结构的操作来实现，例如借助 Sherman-Morrison-Woodbury 公式进行矩阵求逆。本文考虑矩阵平方根和逆平方根运算。给定矩阵的低秩扰动，我们论证了对（逆）平方根存在低秩近似校正。为此，我们建立了真实校正特征值的几何衰减界。随后，我们将该校正表述为一个代数 Riccati 方程的解，并讨论如何计算该方程的低秩解。我们分析了近似求解代数 Riccati 方程时产生的近似误差，给出了谱范数和 Frobenius 范数下的前向与后向误差界。最后，我们描述了算法的若干应用，并通过数值实验证明其实用性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
