Robust Covariance Estimators Based on Information Divergences and Riemannian Manifold

Area of Science:

• Statistics
• Signal Processing
• Matrix Analysis

Background:

• Covariance estimation is crucial in signal processing, especially in heterogeneous environments where data distributions vary.
• Traditional methods often assume data homogeneity or require full knowledge of probability distributions.
• Estimating covariance matrices accurately is challenging under these conditions.

Purpose of the Study:

• To propose a new class of covariance estimators leveraging information divergences.
• To reformulate covariance estimation on the Riemannian manifold of Hermitian positive-definite (HPD) matrices.
• To analyze the robustness of the proposed estimators.

Main Methods:

• Information divergences are used to derive mean estimators on the Riemannian manifold of HPD matrices.
• The geometric properties of the HPD matrix manifold are utilized for mean estimation.
• Robustness is assessed using the influence function analysis.

Main Results:

• The proposed estimators effectively handle heterogeneous environments without needing complete probability distribution knowledge.
• The geometric approach on the HPD manifold provides a robust framework for mean estimation.
• Simulation results demonstrate superior performance and robustness compared to existing covariance estimators.

Conclusions:

• The developed information divergence-based covariance estimators offer a robust and superior alternative for heterogeneous environments.
• The Riemannian manifold approach provides a powerful geometric perspective for statistical estimation problems.
• The findings have implications for advanced signal processing and statistical analysis in complex data settings.