Global Convergence of a Grassmannian Gradient Descent Algorithm for Subspace Estimation
This work addresses subspace estimation for streaming data analysis, offering incremental improvements in convergence guarantees for a non-convex optimization problem.
The paper tackles the problem of estimating a subspace from streaming data using a Grassmannian gradient descent algorithm, proposing an adaptive step size scheme that ensures global convergence to the global minimum with noise-free data and provides an expected convergence rate for noisy data.
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.