论文标题
样品协方差矩阵上的共轭梯度和微小算法的通用性
Universality for the conjugate gradient and MINRES algorithms on sample covariance matrices
论文作者
论文摘要
我们对两种用于求解线性系统的Krylov子空间方法进行了概率分析。我们证明，当应用于满足某些标准力矩条件的宽类样品协方差矩阵时，将偶联梯度和微小算法产生的残留矢量规范是一个中心限制定理。证明涉及为所谓的光谱度量制定四动作用定理，特别是暗示着兰开斯迭代产生的基质的通用性。然后，中央限制定理意味着所讨论的迭代方法几乎确定的迭代计数。
We present a probabilistic analysis of two Krylov subspace methods for solving linear systems. We prove a central limit theorem for norms of the residual vectors that are produced by the conjugate gradient and MINRES algorithms when applied to a wide class of sample covariance matrices satisfying some standard moment conditions. The proof involves establishing a four moment theorem for the so-called spectral measure, implying, in particular, universality for the matrix produced by the Lanczos iteration. The central limit theorem then implies an almost-deterministic iteration count for the iterative methods in question.
