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In this work we consider the problem of estimating a high-dimensional $p\times p$ covariance matrix $\Sigma$, given $n$ observations of confounded datawith covariance $\Sigma + \Gamma \Gamma^T$, where $\Gamma$ is an unknown $p\times q$ matrix of latent factor loadings. We propose a simple and scalableestimator based on the projection on to the right singular vectors of theobserved data matrix, which we call RSVP. Our theoretical analysis of thismethod reveals that in contrast to PCA-based approaches, RSVP is able to copewell with settings where the smallest eigenvalue of $\Gamma^T \Gamma$ is closeto the largest eigenvalue of $\Sigma$, as well as settings where theeigenvalues of $\Gamma^T \Gamma$ are diverging fast. It is also able to handledata that may have heavy tails and only requires that the data has anelliptical distribution. RSVP does not require knowledge or estimation of thenumber of latent factors $q$, but only recovers $\Sigma$ up to an unknownpositive scale factor. We argue this suffices in many applications, for exampleif an estimate of the correlation matrix is desired. We also show that by usingsubsampling, we can further improve the performance of the method. Wedemonstrate the favourable performance of RSVP through simulation experimentsand an analysis of gene expression datasets collated by the GTEX consortium.

RSVP-graphs: Fast High-dimensional Covariance Matrix Estimation under Latent Confounding