Group Symmetric Covariance Estimation

The heavy tails and presence of outliers in real-world datasets motivated the launch of Robust Statistics in the early 60s. We focus on robust covariance matrix estimation and specifically on the M-estimator introduced by D. Tyler in mid-80s. Tyler showed that his estimator is an MLE of a certain population with a non-convex negative log-likelihood, thus making imposition of convex structure, necessary in high-dimensional regime, extremely challenging. Recently, Tyler's target was shown to become convex under a certain change of metric (geodesic convexity). In this work we concentrate on the Group Symmetry (GS) structure meaning that the true covariance matrix commutes with a group of unitary matrices. In engineering applications such structures appear due to natural symmetries of physical processes, e.g. in circulant matrices. GS constraints are convex in the Euclidean metric. We show that they are also convex in the geodesic metric, which enables us to develop GS versions of the sample covariance and Tyler's estimator. Classical results claim that at least n = p and n = p+1 samples are necessary to ensure the existence and uniqueness of the sample covariance and Tyler's estimator, respectively. We significantly improve these bounds under GS conditions and show that in some cases even 1 or 2 samples are sufficient.
Ilya Soloveychik received his BSc degree in Applied Mathematics and Physics from the Moscow Institute of Physics and Technology, Moscow, Russia in 2007, the MSc degree in Mathematics and the PhD degree in Electrical Engineering from the Hebrew University of Jerusalem, Israel in 2013 and 2016, respectively. He is currently a Fulbright postdoctoral fellow with the Harvard School of Engineering and Applied Sciences. His research interests include random matrix theory, high-dimensional statistics and signal processing, and graphical models. He received the Potanin Scholarship for excellence in studies in 2005, the Klein Prize and the Kaete Klausner Scholarship in 2011. In 2015 he was awarded the Feder Family Prize for outstanding research in the field of Communications Technology and in 2016 – the Wolf's Foundation Prize for excellence in studies.