Cleaning large correlation matrices: tools from random matrix theory

Understand

This review covers recent results concerning the estimation of large covariance matrices using tools from Random Matrix Theory (RMT).

• We introduce several RMT methods and analytical techniques, such as the Replica formalism and Free Probability, with an emphasis on the Marchenko-Pastur equation that provides information on the resolvent of multiplicatively corrupted noisy matrices.
• Special care is devoted to the statistics of the eigenvectors of the empirical correlation matrix, which turn out to be crucial for many applications.
• We show in particular how these results can be used to build consistent "Rotationally Invariant" estimators (RIE) for large correlation matrices when there is no prior on the structure of the underlying process.