ASYMMETRY HELPS: EIGENVALUE AND EIGENVECTOR ANALYSES OF ASYMMETRICALLY PERTURBED LOW-RANK MATRICES

Area of Science:

• Statistics
• Linear Algebra
• Machine Learning

Background:

• Estimating symmetric matrices from noisy, asymmetric observations is challenging.
• Standard spectral methods can be biased by asymmetric noise.

Purpose of the Study:

• To investigate the benefits of statistical asymmetry in spectral methods for matrix estimation.
• To develop methods for accurately estimating eigenvalues and eigenvectors of symmetric matrices from asymmetric noise.

Main Methods:

• Utilizing eigen-decomposition on asymmetrically perturbed matrices.
• Developing non-asymptotic eigenvector perturbation bounds.

Main Results:

• The leading eigenvalue estimation accuracy can significantly outperform standard singular value methods.
• The eigen-decomposition approach is adaptive to noise heteroscedasticity without prior knowledge.
• Perturbation bounds for eigenvectors are established, including entrywise bounds.

Conclusions:

• Arranging data samples asymmetrically and applying eigen-decomposition can be highly beneficial for matrix estimation.
• This method offers improved accuracy and robustness compared to traditional techniques.