Semi-Riemannian Manifold Optimization
For researchers in optimization and geometry, this work provides a more general framework that may reduce the need for positive definiteness in metric tensors, but the practical impact is incremental as it extends existing concepts.
The paper introduces a manifold optimization framework using semi-Riemannian structures, which allow indefinite metric tensors, and shows that this weaker geometry suffices for optimizing smooth functions on smooth manifolds, generalizing Riemannian optimization.
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.