Convergence of Sample Eigenvalues, Eigenvectors, and Principal Component Scores for Ultra-High Dimensional Data

Area of Science:

• Biostatistics
• Computational Biology
• Data Science

Background:

• High-throughput biomedical technologies generate high-dimensional datasets.
• Analyzing data where features vastly outnumber samples is a significant challenge.
• Existing methods often address finite or high-dimension low-sample size (HDLSS) regimes separately.

Purpose of the Study:

• To investigate principal component analysis (PCA) in the ultra-high dimensional setting.
• To develop a unified framework that bridges finite and HDLSS regimes.
• To demonstrate the applicability of finite regime results in more general scenarios.

Main Methods:

• Theoretical investigation of principal component analysis (PCA).
• Development of a generalized mathematical framework.
• Numerical simulations to validate theoretical findings.

Main Results:

• Established a unified theoretical framework for PCA under ultra-high dimensions.
• Demonstrated that existing finite-regime results can be generalized.
• Numerical evidence supports the universal applicability of finite regime findings.

Conclusions:

• The proposed framework provides a comprehensive approach to PCA in high-dimensional data.
• This work unifies and extends existing theories for analyzing complex biomedical datasets.
• The findings offer practical implications for statistical analysis in genomics and other high-throughput fields.