A Linearly Convergent Algorithm for Distributed Principal Component Analysis

TL;DR

This paper introduces a neural network-based distributed PCA algorithm that efficiently estimates eigenvectors across multiple machines, guarantees linear convergence, and reduces communication overhead in distributed data settings.

Contribution

It presents the Distributed Sanger's Algorithm, a novel one-time-scale neural network method for distributed PCA with proven linear convergence guarantees.

Findings

Converges linearly to a neighborhood of the true eigenvectors.

Reduces communication overhead compared to existing methods.

Demonstrates effectiveness through numerical experiments.

Abstract

Principal Component Analysis (PCA) is the workhorse tool for dimensionality reduction in this era of big data. While often overlooked, the purpose of PCA is not only to reduce data dimensionality, but also to yield features that are uncorrelated. Furthermore, the ever-increasing volume of data in the modern world often requires storage of data samples across multiple machines, which precludes the use of centralized PCA algorithms. This paper focuses on the dual objective of PCA, namely, dimensionality reduction and decorrelation of features, but in a distributed setting. This requires estimating the eigenvectors of the data covariance matrix, as opposed to only estimating the subspace spanned by the eigenvectors, when data is distributed across a network of machines. Although a few distributed solutions to the PCA problem have been proposed recently, convergence guarantees and/or communications overhead of these solutions remain a concern. With an eye towards communications efficiency, this paper introduces a feedforward neural network-based one time-scale distributed PCA algorithm termed Distributed Sanger's Algorithm (DSA) that estimates the eigenvectors of the data covariance matrix when data is distributed across an undirected and arbitrarily connected network of machines. Furthermore, the proposed algorithm is shown to converge linearly to a neighborhood of the true solution. Numerical results are also provided to demonstrate the efficacy of the proposed solution.