A Cubic Regularized Newton's Method over Riemannian Manifolds

Junyu Zhang; Shuzhong Zhang

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arXiv · 1805.05565

A Cubic Regularized Newton's Method over Riemannian Manifolds

Junyu Zhang·Shuzhong Zhang

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Abstract

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 $\epsilon$-stationary point within $\mathcal{O}(1/\epsilon^{\frac{3}{2}})$ 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.

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Junyu Zhang, Shuzhong Zhang. 2018-05-15. A Cubic Regularized Newton's Method over Riemannian Manifolds. https://arxiv.org/abs/1805.05565

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