# Semi-Riemannian Manifold Optimization

arXiv: https://arxiv.org/abs/1812.07643 · HTML: https://mlsanity.com/paper/1812.07643/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2018.

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

- [Pseudo-Riemannian Graph Convolutional Networks](https://mlsanity.com/paper/2106.03134/) (2021)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [Directed Graph Embeddings in Pseudo-Riemannian Manifolds](https://mlsanity.com/paper/2106.08678/) (2021)
- [Pseudo-Riemannian Embedding Models for Multi-Relational Graph Representations](https://mlsanity.com/paper/2212.03720/) (2022)

## Abstract (arXiv)

We introduce in this paper a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds. Unlike in Riemannian geometry, where each tangent space is equipped with a positive definite inner product, a semi-Riemannian manifold allows the metric tensor to be indefinite on each tangent space, i.e., possessing both positive and negative definite subspaces; differential geometric objects such as geodesics and parallel-transport can be defined on non-degenerate semi-Riemannian manifolds as well, and can be carefully leveraged to adapt Riemannian optimization algorithms to the semi-Riemannian setting. In particular, we discuss the metric independence of manifold optimization algorithms, and illustrate that the weaker but more general semi-Riemannian geometry often suffices for the purpose of optimizing smooth functions on smooth manifolds in practice.