Robust estimation of high-dimensional covariance and precision matrices.

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• High-dimensional data often exhibit complex structures and leptokurtosis.
• Traditional covariance and precision matrix estimators rely on sub-Gaussian assumptions, limiting their applicability.
• Robust estimation methods are needed for broader data distribution classes.

Purpose of the Study:

• To develop robust matrix estimators for high-dimensional data.
• To guarantee estimator performance for distributions beyond sub-Gaussianity.
• To establish theoretical guarantees for the proposed estimators.

Main Methods:

• Development of novel robust matrix estimators.
• Theoretical analysis under bounded fourth moment assumptions.
• Consistency analysis using bounded 2 + ε moments.

Main Results:

• Proposed estimators achieve minimax convergence rates comparable to existing methods under sub-Gaussianity, but with weaker moment assumptions.
• Guaranteed performance for a richer class of distributions.
• Consistency established under a weak bounded moment condition.

Conclusions:

• The developed robust matrix estimators offer improved reliability for complex, high-dimensional datasets.
• These estimators provide theoretical guarantees under weaker distributional assumptions than previously available.
• The findings advance the field of robust statistical inference in high dimensions.