On the matrix square root via geometric optimization

This paper is triggered by the preprint "\emph{Computing Matrix Squareroot via Non Convex Local Search}" by Jain et al. (\textit{\textcolor{blue}{arXiv:1507.05854}}), which analyzes gradient-descent for computing the square root of a positive definite matrix. Contrary to claims of~\citet{jain2015}, our experiments reveal that Newton-like methods compute matrix square roots rapidly and reliably, even for highly ill-conditioned matrices and without requiring commutativity. We observe that gradient-descent converges very slowly primarily due to tiny step-sizes and ill-conditioning. We derive an alternative first-order method based on geodesic convexity: our method admits a transparent convergence analysis ($< 1$ page), attains linear rate, and displays reliable convergence even for rank deficient problems. Though superior to gradient-descent, ultimately our method is also outperformed by a well-known scaled Newton method. Nevertheless, the primary value of our work is its conceptual value: it shows that for deriving gradient based methods for the matrix square root, \emph{the manifold geometric view of positive definite matrices can be much more advantageous than the Euclidean view}.Comment: 8 pages, 12 plots, this version contains several more references and more words about the rank-deficient cas

Provable Efficient Online Matrix Completion via Non-convex Stochastic Gradient Descent

Abstract Matrix completion, where we wish to recover a low rank matrix by observing a few entries from it, is a widely studied problem in both theory and practice with wide applications. Most of the provable algorithms so far on this problem have been restricted to the offline setting where they provide an estimate of the unknown matrix using all observations simultaneously. However, in many applications, the online version, where we observe one entry at a time and dynamically update our estimate, is more appealing. While existing algorithms are efficient for the offline setting, they could be highly inefficient for the online setting. In this paper, we propose the first provable, efficient online algorithm for matrix completion. Our algorithm starts from an initial estimate of the matrix and then performs non-convex stochastic gradient descent (SGD). After every observation, it performs a fast update involving only one row of two tall matrices, giving near linear total runtime. Our algorithm can be naturally used in the offline setting as well, where it gives competitive sample complexity and runtime to state of the art algorithms. Our proofs introduce a general framework to show that SGD updates tend to stay away from saddle surfaces and could be of broader interests to other non-convex problems