High-dimensional principal component analysis with heterogeneous missingness.

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• High-dimensional Principal Component Analysis (PCA) is crucial for dimensionality reduction.
• Missing observations pose a significant challenge in PCA, degrading estimator performance.
• Existing methods like observed-proportion weighted (OPW) estimators struggle with heterogeneous missingness.

Purpose of the Study:

• To develop a robust method for high-dimensional PCA that effectively handles heterogeneous missing data.
• To improve the accuracy and reliability of principal component estimation in the presence of missing observations.
• To address the limitations of current estimators, particularly in realistic, non-uniform missing data scenarios.

Main Methods:

• Introduced primePCA, an iterative imputation and singular space estimation method.
• Leveraged the observed-proportion weighted (OPW) estimator as a starting point.
• Utilized projection and singular value decomposition on imputed data matrices.

Main Results:

• primePCA demonstrates geometric rate of convergence in noiseless cases with sufficient signal strength.
• The method shows improved empirical performance over OPW, especially with heterogeneous missing data.
• Theoretical guarantees depend on average missingness properties, not worst-case scenarios.

Conclusions:

• primePCA offers a significant advancement for high-dimensional PCA with missing data, particularly in heterogeneous settings.
• The method provides accurate principal component recovery where previous approaches failed.
• Numerical studies confirm primePCA's effectiveness on both simulated and real-world datasets, even when data are not Missing Completely At Random.