# Continuation methods for Riemannian Optimization

arXiv: https://arxiv.org/abs/2106.08839 · HTML: https://mlsanity.com/paper/2106.08839/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2021.

## Built on by (most cited first, 1 of 1)

- [Annealed Langevin Posterior Sampling (ALPS): A Rapid Algorithm for Image Restoration with Multiscale Energy Models](https://mlsanity.com/paper/2601.02594/) (2026)

## Abstract (arXiv)

Numerical continuation in the context of optimization can be used to mitigate convergence issues due to a poor initial guess. In this work, we extend this idea to Riemannian optimization problems, that is, the minimization of a target function on a Riemannian manifold. For this purpose, a suitable homotopy is constructed between the original problem and a problem that admits an easy solution. We develop and analyze a path-following numerical continuation algorithm on manifolds for solving the resulting parameter-dependent equation. To illustrate our developments, we consider two typical classical applications of Riemannian optimization: the computation of the Karcher mean and low-rank matrix completion. We demonstrate that numerical continuation can yield improvements for challenging instances of both problems.