Estimating the principal components of correlation matrices from all their empirical eigenvectors
1 Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, associé au CNRS et à l'Université Pierre et Marie Curie - 24 rue Lhomond, 75005 Paris, France
2 Institut de Physique Théorique Philippe Meyer - 24 rue Lhomond, 75005 Paris, France
Received: 8 September 2015
Accepted: 24 November 2015
We consider the problem of estimating the principal components of a population covariance matrix from a limited number of measurement data. Using a combination of random matrix and information-theoretic tools, we show that all the eigenmodes of the sample correlation matrices are informative, and not only the top ones. We show how this information can be exploited when prior information about the principal component, such as whether it is localized or not, is available by mapping the estimation problem onto the search for the ground state of a spin-glass–like effective Hamiltonian encoding the prior. Results are illustrated numerically on the spiked covariance model.
PACS: 02.50.Sk – Multivariate analysis / 02.50.Tt – Inference methods / 05.10.Ln – Monte Carlo methods
© EPLA, 2015