🤖 AI Summary
This study addresses the existence, uniqueness, and algorithmic convergence of covariance matrix estimators in penalized multivariate divergence-based M-estimation under structural constraints. By constructing a regularized M-estimation framework with geodesically convex (not necessarily smooth) penalty functions, the authors rigorously establish the well-posedness of the resulting estimator. They further demonstrate, for the first time, that the standard fixed-point algorithm may fail in this setting and propose a novel reweighted algorithm that integrates tools from geodesic convex analysis and nonsmooth optimization. This new method guarantees monotone convergence for a broad class of penalized M-estimation problems involving structured covariance matrices.
📝 Abstract
In this paper, we study properties of penalized and structured M-estimators of multivariate scatter, based on geodesically convex but not necessarily smooth penalty functions. Existence and uniqueness conditions for these penalized and structured estimators are given. However, we show that the standard fixed-point algorithm which is usually applied to an M-estimation problem does not necessarily converge for penalized M-estimation problems. Hence, we develop a new but simple re-weighting algorithm and prove that it has monotone convergence for a broad class of penalized and structured M-estimators of multivariate scatter.