The conjugate gradient algorithm on a general class of spiked covariance matrices
The conjugate gradient algorithm on a general class of spiked covariance matrices
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DOI：
10.1090/qam/1605
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发表时间：
2021-06
期刊：
影响因子：
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通讯作者：
Xiucai Ding;T. Trogdon
中科院分区：
文献类型：
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作者：
Xiucai Ding;T. Trogdon
我们考虑共轭梯度算法应用于一般类的加标样本协方差矩阵。本文的主要结果是，在任何有限步的误差和残余向量的规范集中在确定性的值由正交多项式确定的变形Marchenko-Pastur法律。一阶极限和涨落被证明是普遍的。此外，对于散装特征值位于一个单一的间隔的情况下，我们表现出更强的普适性的结果，共轭梯度算法的收敛的渐近速度只取决于支持的散装，提供的尖峰是很好地分离的散装。特别是，这表明共轭梯度算法的经典条件数界对于尖峰矩阵是悲观的。
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.