Distributed Estimation of Principal Eigenspaces

TL;DR

This paper introduces a distributed PCA algorithm that efficiently computes principal eigenspaces across multiple machines, analyzing its bias, variance, and convergence properties, and demonstrating its effectiveness compared to centralized PCA.

Contribution

It proposes a distributed PCA method with theoretical analysis of bias, variance, and convergence, applicable even in heterogeneous data settings, matching centralized PCA performance under certain conditions.

Findings

Distributed PCA is unbiased for symmetric innovation distributions.

Convergence rate depends on effective rank, eigen-gap, and number of machines.

Distributed PCA performs comparably to centralized PCA when the number of machines is moderate.

Abstract

Principal component analysis (PCA) is fundamental to statistical machine learning. It extracts latent principal factors that contribute to the most variation of the data. When data are stored across multiple machines, however, communication cost can prohibit the computation of PCA in a central location and distributed algorithms for PCA are thus needed. This paper proposes and studies a distributed PCA algorithm: each node machine computes the top $K$ eigenvectors and transmits them to the central server; the central server then aggregates the information from all the node machines and conducts a PCA based on the aggregated information. We investigate the bias and variance for the resulting distributed estimator of the top $K$ eigenvectors. In particular, we show that for distributions with symmetric innovation, the empirical top eigenspaces are unbiased and hence the distributed PCA is "unbiased". We derive the rate of convergence for distributed PCA estimators, which depends explicitly on the effective rank of covariance, eigen-gap, and the number of machines. We show that when the number of machines is not unreasonably large, the distributed PCA performs as well as the whole sample PCA, even without full access of whole data. The theoretical results are verified by an extensive simulation study. We also extend our analysis to the heterogeneous case where the population covariance matrices are different across local machines but share similar top eigen-structures.