A Cubic Regularized Newton's Method over Riemannian Manifolds

In this paper we present a cubic regularized Newton’s method to minimize a smooth function over a Riemannian manifold. The proposed algorithm is shown to reach a second-order $ϵ$-stationary point within $\mathcal{O}(1\mathrm{/}{ϵ}^{\frac{3}{2}})$ iterations, under the condition that the pullbacks are locally Lipschitz continuous, a condition that is shown to be satisfied if the manifold is compact. Furthermore, we present a local superlinear convergence result under some additional conditions.