# A Riemannian Proximal Newton Method

arXiv: https://arxiv.org/abs/2304.04032 · HTML: https://mlsanity.com/paper/2304.04032/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Apr 2023.

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

- [Causal Abstraction Learning based on the Semantic Embedding Principle](https://mlsanity.com/paper/2502.00407/) (2025)
- [Large-Scale Riemannian Meta-Optimization via Subspace Adaptation](https://mlsanity.com/paper/2501.15235/) (2025)
- [Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances](https://mlsanity.com/paper/2503.02494/) (2025)
- [Robust Non-negative Proximal Gradient Algorithm for Inverse Problems](https://mlsanity.com/paper/2510.23362/) (2025)

## Abstract (arXiv)

In recent years, the proximal gradient method and its variants have been generalized to Riemannian manifolds for solving optimization problems with an additively separable structure, i.e., f + h, where f is continuously differentiable, and h may be nonsmooth but convex with computationally reasonable proximal mapping. In this paper, we generalize the proximal Newton method to embedded submanifolds for solving the type of problem with h(x) = μ |x|1. The generalization relies on the Weingarten and semismooth analysis. It is shown that the Riemannian proximal Newton method has a local quadratic convergence rate under certain reasonable assumptions. Moreover, a hybrid version is given by concatenating a Riemannian proximal gradient method and the Riemannian proximal Newton method. It is shown that if the switch parameter is chosen appropriately, then the hybrid method converges globally and also has a local quadratic convergence rate. Numerical experiments on random and synthetic data are used to demonstrate the performance of the proposed methods.