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

Rajen D. Shah, Benjamin Frot, Gian-Andrea Thanei, Nicolai Meinshausen

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2020 · Volume 82 · Issue 2

Abstract

We consider the problem of estimating a high dimensionalp×pcovariance matrix Σ, givennobservations of confounded data with covariance, where Γ is an unknownp×qmatrix 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 ofis relatively close to the largest eigenvalue of Σ, as well as when the eigenvalues ofare diverging fast. RSVP does not require knowledge or estimation of the number of latent factorsq, but it recovers Σ 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.

Keywords

Causal structure learning, Covariance matrix, Graphical models, High dimensional data, Latent confounding

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