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Asymptotic Joint Distribution of Extreme Sample Eigenvalues and Eigenvectors in the Spiked Population Model
Probability · Submitted 22 Apr 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim As $N, p\to\infty$ but $N/p \to \gamma\in(1, \infty)$, for the sample covariance matrix $S_N := X_NX_N^T/N$, we establish the joint distribution of the largest and the smallest few packs of eigenvalues.
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In this paper, we consider a data matrix $X_N\in\mathbb{R}^{N\times p}$ where all the rows are i.i.d. samples in $\mathbb{R}^p$ of mean zero and covariance matrix $\Sigma\in\mathbb{R}^{p\times p}$. Here the population matrix $\Sigma$ is of finite rank perturbation of the identity matrix. This is the "spiked population model" first proposed by Johnstone. As $N, p\to\infty$ but $N/p \to \gamma\in(1, \infty)$, for the sample covariance matrix $S_N := X_NX_N^T/N$, we establish the joint distribution of the largest and the smallest few packs of eigenvalues. Inside each pack, they will behave the same as the eigenvalues drawn from a Gaussian matrix of the corresponding size. Among different packs, we also calculate the covariance between the Gaussian matrices entries. As a corollary, if all the rows of the data matrix are Gaussian, then these packs will be asymptotically independent. Also, the asymptotic behavior of sample eigenvectors are obtained. Their local fluctuation is also Gaussian with covariance explicitly calculated.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we consider a data matrix $X_N\in\mathbb{R}^{N\times p}$ where all the rows are i.i.d. samples in $\mathbb{R}^p$ of mean zero and covariance matrix $\Sigma\in\mathbb{R}^{p\times p}$. | background 0.46 | 0.27 | 0.02 | asserted 0.96 |
| s1 | Here the population matrix $\Sigma$ is of finite rank perturbation of the identity matrix. | background 0.51 | 0.14 | 0.02 | asserted 1.00 |
| s2 | This is the "spiked population model" first proposed by Johnstone. | background 1.00 | 0.03 | 0.02 | asserted 0.99 |
| s3 | As $N, p\to\infty$ but $N/p \to \gamma\in(1, \infty)$, for the sample covariance matrix $S_N := X_NX_N^T/N$, we establish the joint distribution of the largest and the smallest few packs of eigenvalues. | contribution 0.58 | 0.95 | 0.67 | asserted 1.00 |
| s4 | Inside each pack, they will behave the same as the eigenvalues drawn from a Gaussian matrix of the corresponding size. | result 1.00 | 0.83 | 0.21 | asserted 0.96 |
| s5 | Among different packs, we also calculate the covariance between the Gaussian matrices entries. | result 0.89 | 0.92 | 0.09 | asserted 1.00 |
| s6 | As a corollary, if all the rows of the data matrix are Gaussian, then these packs will be asymptotically independent. | result 0.92 | 0.84 | 0.10 | asserted 1.00 |
| s7 | Also, the asymptotic behavior of sample eigenvectors are obtained. | result 0.68 | 0.91 | 0.10 | asserted 0.99 |
| s8 | Their local fluctuation is also Gaussian with covariance explicitly calculated. | result 1.00 | 0.90 | 0.09 | asserted 1.00 |
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Original abstract
In this paper, we consider a data matrix $X_N\in\mathbb{R}^{N\times p}$ where all the rows are i.i.d. samples in $\mathbb{R}^p$ of mean zero and covariance matrix $\Sigma\in\mathbb{R}^{p\times p}$. Here the population matrix $\Sigma$ is of finite rank perturbation of the identity matrix. This is the "spiked population model" first proposed by Johnstone. As $N, p\to\infty$ but $N/p \to \gamma\in(1, \infty)$, for the sample covariance matrix $S_N := X_NX_N^T/N$, we establish the joint distribution of the largest and the smallest few packs of eigenvalues. Inside each pack, they will behave the same as the eigenvalues drawn from a Gaussian matrix of the corresponding size. Among different packs, we also calculate the covariance between the Gaussian matrices entries. As a corollary, if all the rows of the data matrix are Gaussian, then these packs will be asymptotically independent. Also, the asymptotic behavior of sample eigenvectors are obtained. Their local fluctuation is also Gaussian with covariance explicitly calculated.
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These fluctuations are now well understood (see for example [ref] ).
The Optimal Hard Threshold for Singular Values is 4/sqrt(3) · Conclusion · 0.81
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