Tyler's Covariance Matrix Estimator in Elliptical Models with Convex Structure
This work addresses covariance estimation for elliptical models with known convex structures, offering an incremental improvement in robust statistical methods for applications like signal processing or finance.
The paper tackles structured covariance estimation in elliptical distributions by proposing a convex relaxation (COCA) of the non-convex GMM optimization for Tyler's robust estimator under convex constraints, proving tightness in certain cases and showing in synthetic simulations that COCA outperforms existing methods like Tyler's estimator and its projection.
We address structured covariance estimation in elliptical distributions by assuming that the covariance is a priori known to belong to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization applied to robust Tyler's scatter M-estimator subject to these convex constraints. Unfortunately, GMM turns out to be non-convex due to the objective. Instead, we propose a new COCA estimator - a convex relaxation which can be efficiently solved. We prove that the relaxation is tight in the unconstrained case for a finite number of samples, and in the constrained case asymptotically. We then illustrate the advantages of COCA in synthetic simulations with structured compound Gaussian distributions. In these examples, COCA outperforms competing methods such as Tyler's estimator and its projection onto the structure set.