We consider the conjugate gradient algorithm applied to a general class of spiked sample covariance matrices. The main result of the paper is that the norms of the error and residual vectors at any finite step concentrate on deterministic values determined by orthogonal polynomials with respect to a deformed Marchenko--Pastur law. The first-order limits and fluctuations are shown to be universal. Additionally, for the case where the bulk eigenvalues lie in a single interval we show a stronger universality result in that the asymptotic rate of convergence of the conjugate gradient algorithm only depends on the support of the bulk, provided the spikes are well-separated from the bulk. In particular, this shows that the classical condition number bound for the conjugate gradient algorithm is pessimistic for spiked matrices.

翻译：我们认为,同值梯度算法适用于一整类加压样本共变矩阵。本文的主要结果是,任何有限步骤的误差和残余矢量的规范都集中在由正正方数多角度法确定的确定值上,与分解的马chenko-Pastur法有关。第一阶限值和波动被证明是普遍的。此外,对于大宗电子元值处于单一间隔期的情况,我们显示出一个较强的普遍性结果,即同值梯度算法的缓慢趋同率仅取决于大宗的支撑,只要峰值与大宗完全分离。这特别表明,同值梯度算法的典型条件号对于加压矩阵来说是悲观的。