On sample eigenvalues in a generalized spiked population model

By Z. Bai, Jianfeng Yao. 2008

181 citations recorded by the source corpus.

In the spiked population model introduced by Johnstone (2001) [11], the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein (2006) [5] establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. In a recent work Bai and Yao (2008) [4], we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a generalized spiked population model where the base population covariance matrix is arbitrary, instead of the identity matrix as in Johnstone's case. As the limiting spectral distribution is arbitrary here, new mathematical tools, different from those in Baik and Silverstein (2006) [5], are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes.

Subfields: Mathematical Physics, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

1. Xiucai Ding
2. Jun Yin
3. Antti Knowles
4. Z. Bao
5. Horng-Tzer Yau
6. Ke Wang
7. L. Erdős
8. Debashis Paul

In Statistics and Probability

1. Xiucai Ding
2. Jun Yin
3. Z. Bao
4. Antti Knowles
5. Ke Wang
6. Horng-Tzer Yau
7. Alexander Aue
8. Debashis Paul

Citations in the corpus

• The Full Spectrum of Deep Net Hessians At Scale: Dynamics with Sample Size (2018, background): “Based on these findings, a potential theoretical model for the covariance of the gradients is the spiked covariance model [7] or its generalized version [1].”
• Singular vector and singular subspace distribution for the matrix denoising model (2018, background): “The fluctuation of the extreme eigenvalues of various models have been considered in [2, 3, 7, 8, 9, 23, 24, 28, 14, 15, 42, 55, 34, 56, 59, 44].”
• Fundamental limits of detection in the spiked Wigner model (2018, background): “Landmark investigations in this area (Baik et al., 2005; Baik and Silverstein, 2006; P´ech´e, 2006; F´eral and P´ech´e, 2007; Capitaine et al., 2009; Bai and Yao, 2012, 2008) have established the existence of a spectral threshold above which the top eigenvalue detaches from the bulk of eigenvalues…”
• Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, unknown): “Note that motivated by the seminal article of Baik, Ben Arous and Péché [6], previous works are devoted to the study of the outlier eigenvalues of deformed random Hermitian matrix models, see notably [4, 7, 8, 9, 10, 19, 21, 22, 23, 28, 34, 35, 36, 37, 41, 43, 45, 46].”
• Random matrix theory in statistics: A review (2014, background): “In Bai and Yin (1988), it was proved that after suitable normalization, the ESD of S tends to the semicircular law in this setting.”
• Consistency of AIC and BIC in estimating the number of significant components in high-dimensional principal component analysis (2018, background): “For more details, see Bai and Yao (2012) and the references therein.”
• Statistical inference for principal components of spiked covariance matrices (2020, background): “On extreme eigenvalues, further study for more generally distributed covariance matrices can be found in [8, 22, 12, 69, 5, 6, 16, 27, 52].”
• On the principal components of sample covariance matrices (2014, background): “The ratio of the right- and left-sided limits at 2 of ∆ lies in [1, 3].”
• Identifying the number of factors from singular values of a large sample auto-covariance matrix (2014, background): “This is thus much in line with what is known for the phase transition phenomenon for large sample covariance matrices as exposed in [6] and [5].”