High-dimensional covariance estimation from a small number of samples

We synthesize knowledge from numerical weather prediction, inverse theory and statistics to address the problem of estimating a high-dimensional covariance matrix from a small number of samples. This problem is fundamental in statistics, machine learning/artificial intelligence, and in modern Earth science. We create several new adaptive methods for high-dimensional covariance estimation, but one method, which we call NICE (noise-informed covariance estimation), stands out because it has three important properties: (i) NICE is conceptually simple and computationally efficient, (ii) NICE guarantees symmetric positive semi-definite covariance estimates, and (iii) NICE is largely tuning-free. We illustrate the use of NICE on a large set of Earth-science-inspired numerical examples, including cycling data assimilation, geophysical inversion of electromagnetic data, and training of feed-forward neural networks with time-averaged data from a chaotic dynamical system. Our theory, heuristics and numerical tests suggest that NICE may indeed be a viable option for high-dimensional covariance estimation in many Earth science problems. This is joint work with David Vishny (Scripps Institution of Oceanography), Kyle Gwirtz (University of Maryland, Baltimore County and NASA Goddard Space Flight Center), Eviatar Bach (University of Reading), Oliver Dunbar (Caltech) and Daniel Hodyss (Naval Research Laboratory).