The Fast Convergence of Incremental PCA

TL;DR

This paper analyzes the convergence rates of classical incremental PCA algorithms, Krasulina and Oja, providing finite-sample guarantees for estimating the top eigenvector in high-dimensional settings.

Contribution

It offers the first finite-sample convergence analysis for Krasulina and Oja algorithms in the context of incremental PCA.

Findings

Finite-sample convergence rates established for Krasulina and Oja algorithms.

Algorithms maintain O(d) space complexity while updating estimates.

Results demonstrate efficient top eigenvector estimation in high dimensions.

Abstract

We consider a situation in which we see samples in $\mathbb{R}^d$ drawn i.i.d. from some distribution with mean zero and unknown covariance A. We wish to compute the top eigenvector of A in an incremental fashion - with an algorithm that maintains an estimate of the top eigenvector in O(d) space, and incrementally adjusts the estimate with each new data point that arrives. Two classical such schemes are due to Krasulina (1969) and Oja (1983). We give finite-sample convergence rates for both.