A Stochastic PCA and SVD Algorithm with an Exponential Convergence Rate

TL;DR

This paper introduces VR-PCA, a stochastic algorithm for PCA and SVD that achieves exponential convergence with low computational cost, addressing limitations of existing methods.

Contribution

It applies a variance-reduced stochastic gradient technique to PCA and SVD, providing the first exponential convergence analysis for this non-convex problem.

Findings

Achieves exponential convergence rate.

Uses computationally cheap stochastic iterations.

Outperforms existing algorithms in convergence speed.

Abstract

We describe and analyze a simple algorithm for principal component analysis and singular value decomposition, VR-PCA, which uses computationally cheap stochastic iterations, yet converges exponentially fast to the optimal solution. In contrast, existing algorithms suffer either from slow convergence, or computationally intensive iterations whose runtime scales with the data size. The algorithm builds on a recent variance-reduced stochastic gradient technique, which was previously analyzed for strongly convex optimization, whereas here we apply it to an inherently non-convex problem, using a very different analysis.