Semi-Riemannian Manifold Optimization

TL;DR

This paper develops a new optimization framework on semi-Riemannian manifolds, extending Riemannian optimization techniques to indefinite metric spaces, broadening the scope of manifold optimization methods.

Contribution

It introduces a semi-Riemannian manifold optimization framework, enabling the use of indefinite metrics for optimization on smooth manifolds, which was not previously explored.

Findings

Semi-Riemannian geometry can be used for manifold optimization.

Optimization algorithms can be metric-independent in this setting.

The framework generalizes Riemannian optimization methods.

Abstract

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.