ABSTRACT:

These lectures will survey the theoretical foundations of large sample covariance matrices. The goal is a comprehensive picture of the eigenvalues and eigenvectors (for instance the principal components) of sample covariance matrices describing highly correlated populations. After reviewing the basics, an overview of recent developments will be given, this also by paying special attention to the mathematical tools, such as local laws and universality, underlying the proofs.

 

Lecture 1. THE PRINCIPAL COMPONENTS OF SAMPLE COVARIANCE MATRICES

(March 2, 17:30 - 19:00, Lecture Theatre 340)

 In the first lecture I explain how sample covariance matrices arise in the statistical analysis of correlated data. I focus on the principal components, i.e. the top eigenvalues and associated eigenvectors. In the first two lectures, I discuss so-called spiked models, where the population covariance matrix differs from the identity by a matrix of small rank. I give an overview of the main features of the principal components: the BBP phase transition, outlier eigenvalues, non-outlier eigenvalue sticking and Tracy-Widom-Airy statistics, cone concentration of outlier eigenvectors, and a weak bias for the non-outlier eigenvectors.

 Lecture 2. THE KEY TOOLS AND OUTLINE OF PROOFS

 (March 3, 17:30 - 19:00, Clore Lecture Theatre)

In the second lecture I give some ideas how the results from the first lecture can be proved. The main tool is the isotropic Marchenko-Pastur (MP) law. Combined with simple linear algebra, some complex analysis, and bootstrapping arguments, it yields detailed information about the principal components. More refined information about eigenvectors is obtained in conjunction with a probabilistic version of quantum unique ergodicity.

 Lecture 3. BEYOND SPIKED MODELS: ARBITRARY COVARIANCE

 (March 4, 17:30 - 19:00, Lecture Theatre 340)

In the final lecture I explain how to leave the realm of spiked models, and consider data with an arbitrary population covariance matrix. The model is now no longer governed by the MP law, but by the so-called free multiplicative convolution of the MP law with the spectral measure of the population covariance matrix. The isotropic MP law from the previous lecture has to be generalized to an anisotropic law, which may be regarded as a quantitative result from free probability theory. In this general setting, I revisit the features of sample covariance matrices outlined in the first lecture.

 

REFERENCES:
 

1. On the principal components of sample covariance matrices, A. Bloemendal, A. Knowles, H.-T. Yau, and J. Yin. Preprint arXiv:1404.0788.
2.  Isotropic local laws for sample covariance and generalized Wigner matrices, A. Bloemendal, A. Knowles, L. Erdős, H.-T. Yau, and J. Yin. Elect. J. Prob. 19, Article 33, 1-53 (2014).

3. Anisotropic local laws for random matrices. A. Knowles and J. Yin. Preprint arXiv:1410.3516.