Dimension Reduction for Large-Scale Federated Data: Statistical Rate and Asymptotic Inference

Area of Science:

• Statistics
• Machine Learning
• Computational Biology

Background:

• Traditional Principal Component Analysis (PCA) faces challenges with large-scale federated data due to privacy and computational costs.
• Existing distributed algorithms often struggle with both high dimensionality and massive sample sizes simultaneously.

Purpose of the Study:

• To propose a novel distributed PCA method (FADI) for ultra-large dimensional and sample-sized federated data.
• To develop a general framework for statistical problems in distributed settings.
• To analyze the computational efficiency and error rates of the proposed method.

Main Methods:

• Developed FAst DIstributed (FADI) PCA by combining parallel computing along dimensions and distributed computing along samples.
• Utilized L parallel copies of p-dimensional fast sketches to divide computational burden.
• Established a general theoretical framework with comprehensive results for statistical problems.

Main Results:

• FADI accelerates computation while maintaining the same non-asymptotic error rate as traditional PCA when L*p >= d.
• Derived inferential results characterizing the asymptotic distribution of FADI.
• Observed a phase-transition phenomenon as L*p increases.

Conclusions:

• FADI offers an efficient solution for PCA on ultra-large federated datasets.
• The method provides theoretical guarantees on error rates and asymptotic distributions.
• FADI was successfully applied to analyze population structure in the 1000 Genomes data.