On the matrix square root via geometric optimization

This paper is triggered by the preprint "Computing Matrix Squareroot via Non Convex Local Search" by Jain et al. (textittextcolorbluearXiv:1507.05854), which analyzes gradient-descent for computing the square root of a positive definite matrix. Contrary to claims of~citetjain2015, 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, the manifold geometric view of positive definite matrices can be much more advantageous than the Euclidean view.