# A Cubic Regularized Newton's Method over Riemannian Manifolds

arXiv: https://arxiv.org/abs/1805.05565 · HTML: https://mlsanity.com/paper/1805.05565/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2018.

## Built on by (most cited first, 4 of 4)

- [Scalable Second-order Riemannian Optimization for K-means Clustering](https://mlsanity.com/paper/2509.21675/) (2025)
- [Riemannian Stochastic Gradient Method for Nested Composition Optimization](https://mlsanity.com/paper/2207.09350/) (2022)
- [Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness](https://mlsanity.com/paper/2209.01229/) (2022)
- [Faster Riemannian Newton-type Optimization by Subsampling and Cubic Regularization](https://mlsanity.com/paper/2302.11076/) (2023)

## Abstract (arXiv)

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 O(1/ε^frac32) 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.