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 - i.e. the specific threshold it selects gives exactly
the smallest achievable MSE loss among all possible threshold choices for that
noisy dataset 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 model, 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 where there is inter-element 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