Global Convergence of a Grassmannian Gradient Descent Algorithm for Subspace Estimation

TL;DR

This paper proves that a Grassmannian gradient descent algorithm with an adaptive step size converges globally for noiseless subspace estimation and provides convergence rates in noisy scenarios.

Contribution

It introduces an adaptive step size scheme for Grassmannian gradient descent that guarantees global convergence in noiseless cases and analyzes convergence rates with noise.

Findings

Converges from any initialization in noiseless data

Expected convergence rate established for noisy data

Adaptive step size improves convergence performance

Abstract

It has been observed in a variety of contexts that gradient descent methods have great success in solving low-rank matrix factorization problems, despite the relevant problem formulation being non-convex. We tackle a particular instance of this scenario, where we seek the $d$-dimensional subspace spanned by a streaming data matrix. We apply the natural first order incremental gradient descent method, constraining the gradient method to the Grassmannian. In this paper, we propose an adaptive step size scheme that is greedy for the noiseless case, that maximizes the improvement of our metric of convergence at each data index $t$, and yields an expected improvement for the noisy case. We show that, with noise-free data, this method converges from any random initialization to the global minimum of the problem. For noisy data, we provide the expected convergence rate of the proposed algorithm per iteration.