On the principal components of sample covariance matrices

By Alex Bloemendal, Antti Knowles, Horng-Tzer Yau, Jun Yin. 2014

165 citations recorded by the source corpus.

We introduce a class of $$M \times M$$M×M sample covariance matrices $${\mathcal {Q}}$$Q which subsumes and generalizes several previous models. The associated population covariance matrix $$\Sigma = \mathbb {E}{\mathcal {Q}}$$Σ=EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of $$\Sigma - I_M$$Σ-IM may depend on $$M$$M in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of $${\mathcal {Q}}$$Q. We derive precise large deviation estimates on the generalized components $$\langle {\mathbf{{w}}} , {\varvec{\xi }_i}\rangle $$⟨w,ξi⟩ of the outlier and non-outlier eigenvectors $$\varvec{\xi }_i$$ξi. Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of $$\Sigma $$Σ. The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix $${\mathcal {Q}}$$Q, satisfying $$\mathbb {E}{\mathcal {Q}} = I_M$$EQ=IM, as input: the isotropic local Marchenko–Pastur law established in Bloemendal et al. (Electron J Probab 19:1–53, 2014), level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue–eigenvector distribution.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/686affb49c4daa96d8060fda3ac359730dfa3115

How researchers see this paper

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

In Condensed Matter Physics

1. Xiucai Ding
2. L. Erdős
3. Fan Yang
4. Horng-Tzer Yau
5. Tommaso Cai
6. Z. Bao
7. Jiaoyang Huang
8. Arka Adhikari

In Mathematical Physics

1. Z. Bao
2. Xiucai Ding
3. Guangming Pan
4. L. Erdős
5. Jun Yin
6. Fan Yang
7. Ke Wang
8. Horng-Tzer Yau

In Statistics and Probability

1. Xiucai Ding
2. Z. Bao
3. Fan Yang
4. Guangming Pan
5. Ke Wang
6. Jun Yin
7. L. Erdős
8. Horng-Tzer Yau

Citations in the corpus

• Singular vector and singular subspace distribution for the matrix denoising model (2018, background): “By establishing a general universality result of the eigenvectors of the sample covariance matrix in the null case, the authors of [13] are able to show that the law of the eigenvectors of the spiked covariance matrices are asymptotically Gaussian in the subcritical regime.”
• Spiked separable covariance matrices and principal components (2019, methodology): “Hence as in [8], we shall consider the generalized components xv,PS wy of the random projection PS :“ ÿ”
• A Dynamical Approach to Random Matrix Theory (2017, methodology): “For applications in principal component analysis, the main focus is on the extreme eigenvalues and the outliers whose detailed analysis was given in [22].”
• Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices (2017, methodology): “[13] obtained the precise large deviation of the spiked eigenvalues and non-spiked eigenvalues under a more general model than (1.”