ScreeNOT: Exact MSE-optimal singular value thresholding in correlated noise

Abstract

We derive a formula for optimal hard thresholding of the singular value decomposition in the presence of correlated additive noise; although it nominally involves unobservables, we show how to apply it even where the noise covariance structure is not a priori known or is not independently estimable. The proposed method, which we call ScreeNOT, is a mathematically solid alternative to Cattell’s ever-popular but vague scree plot heuristic from 1966. ScreeNOT has a surprising oracle property: it typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on each given problem instance, that is, the specific threshold it selects gives exactly the smallest achievable MSE loss among all possible threshold choices for that noisy data set and that unknown underlying true low rank model. The method is computationally efficient and robust against perturbations of the underlying covariance structure. Our results depend on the assumption that the singular values of the noise have a limiting empirical distribution of compact support; this property, which is standard in random matrix theory, is satisfied by many models exhibiting either cross-row correlation structure or cross-column correlation structure, and also by many situations with more general, interelement correlation structure. Simulations demonstrate the effectiveness of the method even at moderate matrix sizes. The paper is supplemented by ready-to-use software packages implementing the proposed algorithm: package $\mathtt{ScreeNOT}$ in $\mathtt{Python}$ (via $\mathtt{PyPI}$) and $\mathtt{R}$ (via $\mathtt{CRAN}$).