Spiked multiplicative random matrices and principal components
Spiked multiplicative random matrices and principal components
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DOI：
10.1016/j.spa.2023.05.009
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发表时间：
2023-02
影响因子：
1.4
通讯作者：
Xiucai Ding;H. Ji
中科院分区：
文献类型：
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作者：
Xiucai Ding;H. Ji
在本文中，我们研究了当随机性来自 Haar 矩阵时尖峰不变乘法模型的特征值和特征向量。我们以最佳收敛速度建立离群特征值 λ ̂ i 和离群特征向量 u ̂ i 的广义分量（< v, u ̂ i> 对于任何确定性向量 v）的限制。此外，我们证明非离群特征值与无尖峰矩阵的特征值一致，并且非离群特征向量是离域的。对于简并尖峰，该结果也适用于所谓的 BBP 转变。一方面，我们的结果可以被视为在附加正则条件下对 Belinschi 等人（2017）的对应结果的改进。另一方面，它们可以被视为 Ding 和 Yang (2021) 的类似物，通过用 Haar 随机矩阵替换具有独立同分布项的随机矩阵。
In this paper, we study the eigenvalues and eigenvectors of the spiked invariant multiplicative models when the randomness is from Haar matrices. We establish the limits of the outlier eigenvalues λ ̂ i and the generalized components (< v, u ̂ i> for any deterministic vector v) of the outlier eigenvectors u ̂ i with optimal convergence rates. Moreover, we prove that the non-outlier eigenvalues stick with those of the unspiked matrices and the non-outlier eigenvectors are delocalized. The results also hold near the so-called BBP transition and for degenerate spikes. On one hand, our results can be regarded as a refinement of the counterparts of Belinschi et al.(2017) under additional regularity conditions. On the other hand, they can be viewed as an analog of Ding and Yang (2021) by replacing the random matrix with iid entries with Haar random matrix.