# A new, globally convergent Riemannian conjugate gradient method

arXiv: https://arxiv.org/abs/1302.0125 · HTML: https://mlsanity.com/paper/1302.0125/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2013.

## Built on by (most cited first, 5 of 5)

- [Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold](https://mlsanity.com/paper/2308.10547/) (2023)
- [Riemannian Stochastic Gradient Method for Nested Composition Optimization](https://mlsanity.com/paper/2207.09350/) (2022)
- [Faster Riemannian Newton-type Optimization by Subsampling and Cubic Regularization](https://mlsanity.com/paper/2302.11076/) (2023)
- [Riemannian Zeroth-Order Gradient Estimation with Structure-Preserving Metrics for Geodesically Incomplete Manifolds](https://mlsanity.com/paper/2601.08039/) (2026)
- [Stiefel-AdamW: Geometry-Aware AdamW for Linear Factorization Blocks](https://mlsanity.com/paper/2609.21039/) (2026)

## Abstract (arXiv)

This article deals with the conjugate gradient method on a Riemannian manifold with interest in global convergence analysis. The existing conjugate gradient algorithms on a manifold endowed with a vector transport need the assumption that the vector transport does not increase the norm of tangent vectors, in order to confirm that generated sequences have a global convergence property. In this article, the notion of a scaled vector transport is introduced to improve the algorithm so that the generated sequences may have a global convergence property under a relaxed assumption. In the proposed algorithm, the transported vector is rescaled in case its norm has increased during the transport. The global convergence is theoretically proved and numerically observed with examples. In fact, numerical experiments show that there exist minimization problems for which the existing algorithm generates divergent sequences, but the proposed algorithm generates convergent sequences.