Group Symmetry and non-Gaussian Covariance Estimation

TL;DR

This paper demonstrates that group symmetry constraints in non-Gaussian covariance estimation are geodesically convex, enabling improved estimation accuracy through prior symmetry knowledge, with practical algorithms and experimental validation.

Contribution

It proves the geodesic convexity of symmetry-constrained covariance sets and introduces a numerical method for constrained non-Gaussian covariance estimation.

Findings

Symmetry constraints are geodesically convex.

The proposed method improves estimation accuracy.

Numerical experiments validate performance gains.

Abstract

We consider robust covariance estimation with group symmetry constraints. Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and Multivariate Generalized Gaussian distribution methods, usually involve non-convex minimization problems. Recently, it was shown that the underlying principle behind their success is an extended form of convexity over the geodesics in the manifold of positive definite matrices. A modern approach to improve estimation accuracy is to exploit prior knowledge via additional constraints, e.g., restricting the attention to specific classes of covariances which adhere to prior symmetry structures. In this paper, we prove that such group symmetry constraints are also geodesically convex and can therefore be incorporated into various non-Gaussian covariance estimators. Practical examples of such sets include: circulant, persymmetric and complex/quaternion proper structures. We provide a simple numerical technique for finding maximum likelihood estimates under such constraints, and demonstrate their performance advantage using synthetic experiments.