ENTRYWISE EIGENVECTOR ANALYSIS OF RANDOM MATRICES WITH LOW EXPECTED RANK.

Area of Science:

• Statistical Machine Learning
• Random Matrix Theory
• Data Science

Background:

• Recovering low-rank structures is crucial in machine learning applications like factor analysis and community detection.
• Existing eigenvector error bounds are often insufficient for critical entrywise analyses.

Purpose of the Study:

• To investigate the entrywise behavior of eigenvectors for random matrices with low-rank expectations.
• To settle a conjecture regarding the spectral algorithm's exact recovery in the stochastic block model.

Main Methods:

• Developing a first-order approximation for eigenvectors under the ℓ∞ norm.
• Analyzing the tightness and linearity of the approximation for sharp comparisons between eigenvectors and their expectations.
• Extending results to eigenspace perturbations.

Main Results:

• A novel, tight first-order approximation for eigenvectors under the ℓ∞ norm is established.
• The approximation allows for precise comparisons of eigenvectors and their expectations, even with large perturbations.
• New ℓ∞-type bounds are derived for synchronization and noisy matrix completion.

Conclusions:

• The findings confirm the spectral algorithm's exact recovery capabilities in the stochastic block model without additional steps.
• The developed entrywise analysis provides a powerful tool for understanding and improving low-rank structure recovery in various machine learning tasks.
• This work advances the theoretical understanding of random matrix theory in the context of complex data analysis.