Escaping from saddle points on Riemannian manifolds

We consider minimizing a nonconvex, smooth function f on a Riemannian manifold M. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to escape saddle points on the manifold). The rate of convergence depends as 1/epsilon(2) on the accuracy c, which matches a rate known only for unconstrained smooth minimization. The convergence rate depends polylogarithmically on the manifold dimension d, hence is almost dimension -free. The rate also has a polynomial dependence on the parameters describing the curvature of the manifold and the smoothness of the function. While the unconstrained problem (Euclidean setting) is well -studied, our result is the first to prove such a rate for nonconvex, manifold-constrained problems.