Right singular vector projection graphs: fast high dimensional covariance matrix estimation under latent confounding

We consider the problem of estimating a high dimensional pxp covariance matrix sigma, given n observations of confounded data with covariance sigma+Gamma Gamma T, where Gamma is an unknown pxq matrix of latent factor loadings. We propose a simple and scalable estimator based on the projection onto the right singular vectors of the observed data matrix, which we call right singular vector projection (RSVP). Our theoretical analysis of this method reveals that, in contrast with approaches based on the removal of principal components, RSVP can cope well with settings where the smallest eigenvalue of Gamma T Gamma is relatively close to the largest eigenvalue of sigma, as well as when the eigenvalues of Gamma T Gamma are diverging fast. RSVP does not require knowledge or estimation of the number of latent factors q, but it recovers sigma only up to an unknown positive scale factor. We argue that this suffices in many applications, e.g. if an estimate of the correlation matrix is desired. We also show that, by using subsampling, we can further improve the performance of the method. We demonstrate the favourable performance of RSVP through simulation experiments and an analysis of gene expression data sets collated by the GTEX consortium.