An Eigenvector Perturbation Bound and Its Application to Robust Covariance Estimation.

Area of Science:

• Statistics
• Machine Learning
• Linear Algebra

Background:

• Eigenvectors and singular vectors are crucial in data analysis.
• Matrix perturbations from noise can affect eigenvector accuracy.
• Existing bounds, like the Davis-Kahan sin θ theorem, have limitations.

Purpose of the Study:

• To derive a tighter perturbation bound for singular vectors of low-rank, incoherent matrices.
• To apply this improved bound to robust covariance estimation.
• To develop and analyze new robust covariance estimators for heavy-tailed data.

Main Methods:

• Theoretical analysis of matrix perturbation theory.
• Development of a refined perturbation bound for singular vectors.
• Construction of novel robust covariance estimators.
• Asymptotic analysis of estimator properties.

Main Results:

• A new perturbation bound for singular vectors is established, showing improvement by factors of sqrt(d1) or sqrt(d2) for low-rank, incoherent matrices.
• The new bound enhances the performance of robust covariance estimation, particularly for heavy-tailed distributions.
• Novel robust covariance estimators are proposed with proven asymptotic properties.

Conclusions:

• The refined perturbation bound offers significant advantages for analyzing matrix properties in the presence of noise.
• The proposed robust covariance estimators provide a powerful tool for statistical analysis with heavy-tailed data.
• This work bridges theoretical advancements in perturbation theory with practical applications in robust statistics.